%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.14", %%% date = "16 March 2019", %%% time = "10:40:45 MDT", %%% filename = "ejp.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% FAX = "+1 801 581 4148", %%% URL = "http://www.math.utah.edu/~beebe", %%% checksum = "46158 46879 235002 2207790", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "bibliography; BibTeX; Electronic %%% Journal of Probability", %%% license = "public domain", %%% supported = "yes", %%% docstring = "This is a COMPLETE bibliography of %%% publications in the open-source journal, %%% Electronic Journal of Probability (CODEN %%% none, ISSN 1083-6489, ISSN-L 1083-6489) %%% published in collaboration with the Institute %%% of Mathematical Statistics. Publication %%% began at the University of Washington %%% (Seattle, WA, USA) with volume 1, number 1, %%% in 1996. There is only one volume per year, %%% but articles are available online as soon as %%% they have been accepted for publication. %%% %%% In 2016, journal hosting moved to Project %%% Euclid. %%% %%% The journal has Web sites at %%% %%% https://projecteuclid.org/euclid.ejp %%% http://ejp.ejpecp.org/ %%% http://www.math.washington.edu/~ejpecp/EJP/ %%% %%% There is also a companion journal for shorter %%% communications: it is covered in ecp.bib. %%% %%% At version 1.14, the year coverage looked %%% like this: %%% %%% 1996 ( 14) 2004 ( 29) 2012 ( 107) %%% 1997 ( 9) 2005 ( 46) 2013 ( 109) %%% 1998 ( 16) 2006 ( 50) 2014 ( 122) %%% 1999 ( 23) 2007 ( 58) 2015 ( 129) %%% 2000 ( 14) 2008 ( 76) 2016 ( 70) %%% 2001 ( 32) 2009 ( 94) 2017 ( 97) %%% 2002 ( 16) 2010 ( 73) 2018 ( 120) %%% 2003 ( 23) 2011 ( 92) 2019 ( 17) %%% %%% Article: 1436 %%% %%% Total entries: 1436 %%% %%% Data for this bibliography have been derived %%% primarily from data at the publisher Web %%% site, with contributions from the BibNet %%% Project and TeX User Group bibliography %%% archives, and the MathSciNet and zbMATH %%% databases. %%% %%% Numerous errors in the sources noted above %%% have been corrected. Spelling has been %%% verified with the UNIX spell and GNU ispell %%% programs using the exception dictionary %%% stored in the companion file with extension %%% .sok. %%% %%% BibTeX citation tags are uniformly chosen %%% as name:year:abbrev, where name is the %%% family name of the first author or editor, %%% year is a 4-digit number, and abbrev is a %%% 3-letter condensation of important title %%% words. Citation tags were automatically %%% generated by the biblabel software %%% developed for the BibNet Project. %%% %%% In this bibliography, entries are sorted in %%% publication order, with the help of %%% ``bibsort -bypages''. %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ====================================================================

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%%% ==================================================================== %%% Acknowledgement abbreviations:

@String{ack-nhfb= "Nelson H. F. Beebe, University of Utah, Department of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1 801 581 4148, e-mail: \path|beebe@math.utah.edu|, \path|beebe@acm.org|, \path|beebe@computer.org| (Internet), URL: \path|http://www.math.utah.edu/~beebe/|"}

%%% ==================================================================== %%% Journal abbreviations:

@String{j-ELECTRON-J-PROBAB= "Electronic Journal of Probability"}

%%% ==================================================================== %%% Bibliography entries, sorted in publication order with %%% ``bibsort -byvolume'':

@Article{Khoshnevisan:1996:LCS, author = "Davar Khoshnevisan", title = "{L{\'e}vy} classes and self-normalization", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "1:1--1:18", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-1", ISSN = "1083-6489", MRclass = "60F15 (60J15 60J45 60J55)", MRnumber = "1386293 (97h:60024)", MRreviewer = "Qi Man Shao", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/1; http://www.math.washington.edu/~ejpecp/EjpVol1/paper1.abs.html", abstract = "We prove a Chung's law of the iterated logarithm for recurrent linear Markov processes. In order to attain this level of generality, our normalization is random. In particular, when the Markov process in question is a diffusion, we obtain the integral test corresponding to a law of the iterated logarithm due to Knight.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Self-normalization, Levy Classes", } @Article{Lawler:1996:HDC, author = "Gregory F. Lawler", title = "{Hausdorff} dimension of cut points for {Brownian} motion", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "2:1--2:20", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-2", ISSN = "1083-6489", MRclass = "60J65", MRnumber = "1386294 (97g:60111)", MRreviewer = "Paul McGill", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/2", abstract = "Let $B$ be a Brownian motion in $ R^d$, $ d = 2, 3$. A time $ t \in [0, 1]$ is called a cut time for $ B[0, 1]$ if $ B[0, t) \cap B(t, 1] = \emptyset $. We show that the Hausdorff dimension of the set of cut times equals $ 1 - \zeta $, where $ \zeta = \zeta_d$ is the intersection exponent. The theorem, combined with known estimates on $ \zeta_3$, shows that the percolation dimension of Brownian motion (the minimal Hausdorff dimension of a subpath of a Brownian path) is strictly greater than one in $ R^3$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, Hausdorff dimension, cut points, intersection exponent", } @Article{Bass:1996:EEB, author = "Richard F. Bass and Krzysztof Burdzy", title = "Eigenvalue expansions for {Brownian} motion with an application to occupation times", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "3:1--3:19", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-3", ISSN = "1083-6489", MRclass = "60J65", MRnumber = "1386295 (97c:60201)", MRreviewer = "Zhong Xin Zhao", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/3; http://www.math.washington.edu/~ejpecp/EjpVol1/paper3.abs.html", abstract = "Let $B$ be a Borel subset of $ R^d$ with finite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting $B$. Let $ A_1$ be the time spent by Brownian motion in a closed cone with vertex $0$ until time one. We show that $ \lim_{u \to 0} \log P^0 (A_1 < u) / \log u = 1 / \xi $ where $ \xi $ is defined in terms of the first eigenvalue of the Laplacian in a compact domain. Eigenvalues of the Laplacian in open and closed sets are compared.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, eigenfunction expansion, eigenvalues, arcsine law", } @Article{Pitman:1996:RDD, author = "Jim Pitman and Marc Yor", title = "Random Discrete Distributions Derived from Self-Similar Random Sets", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "4:1--4:28", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-4", ISSN = "1083-6489", MRclass = "60D05", MRnumber = "1386296 (98i:60010)", MRreviewer = "Bert Fristedt", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/4", abstract = "A model is proposed for a decreasing sequence of random variables $ (V_1, V_2, \cdots) $ with $ \sum_n V_n = 1 $, which generalizes the Poisson--Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let $ V_n $ be the length of the $n$ th longest component interval of $ [0, 1] \backslash Z$, where $Z$ is an a.s. non-empty random closed of $ (0, \infty)$ of Lebesgue measure $0$, and $Z$ is self-similar, i.e., $ c Z$ has the same distribution as $Z$ for every $ c > 0$. Then for $ 0 \leq a < b \leq 1$ the expected number of $n$'s such that $ V_n \in (a, b)$ equals $ \int_a^b v^{-1} F(d v)$ where the structural distribution $F$ is identical to the distribution of $ 1 - \sup (Z \cap [0, 1])$. Then $ F(d v) = f(v)d v$ where $ (1 - v) f(v)$ is a decreasing function of $v$, and every such probability distribution $F$ on $ [0, 1]$ can arise from this construction.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "interval partition, zero set, excursion lengths, regenerative set, structural distribution", } @Article{Seppalainen:1996:MMB, author = "Timo Sepp{\"a}l{\"a}inen", title = "A microscopic model for the {Burgers} equation and longest increasing subsequences", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "5:1--5:51", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-5", ISSN = "1083-6489", MRclass = "60K35 (35Q53 60C05 82C22)", MRnumber = "1386297 (97d:60162)", MRreviewer = "Shui Feng", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/5", abstract = "We introduce an interacting random process related to Ulam's problem, or finding the limit of the normalized longest increasing subsequence of a random permutation. The process describes the evolution of a configuration of sticks on the sites of the one-dimensional integer lattice. Our main result is a hydrodynamic scaling limit: The empirical stick profile converges to a weak solution of the inviscid Burgers equation under a scaling of lattice space and time. The stick process is also an alternative view of Hammersley's particle system that Aldous and Diaconis used to give a new solution to Ulam's problem. Along the way to the scaling limit we produce another independent solution to this question. The heart of the proof is that individual paths of the stochastic process evolve under a semigroup action which under the scaling turns into the corresponding action for the Burgers equation, known as the Lax formula. In a separate appendix we use the Lax formula to give an existence and uniqueness proof for scalar conservation laws with initial data given by a Radon measure.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Hydrodynamic scaling limit, Ulam's problem, Hammersley's process, nonlinear conservation law, the Burgers equation, the Lax formula", } @Article{Fleischmann:1996:TSA, author = "Klaus Fleischmann and Andreas Greven", title = "Time-Space Analysis of the Cluster-Formation in Interacting Diffusions", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "6:1--6:46", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-6", ISSN = "1083-6489", MRclass = "60K35 (60J60)", MRnumber = "1386298 (97e:60151)", MRreviewer = "Ingemar Kaj", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/6", abstract = "A countable system of linearly interacting diffusions on the interval [0, 1], indexed by a hierarchical group is investigated. A particular choice of the interactions guarantees that we are in the diffusive clustering regime, that is spatial clusters of components with values all close to 0 or all close to 1 grow in various different scales. We studied this phenomenon in [FG94]. In the present paper we analyze the evolution of single components and of clusters over time. First we focus on the time picture of a single component and find that components close to 0 or close to 1 at a late time have had this property for a large time of random order of magnitude, which nevertheless is small compared with the age of the system. The asymptotic distribution of the suitably scaled duration a component was close to a boundary point is calculated. Second we study the history of spatial 0- or 1-clusters by means of time scaled block averages and time-space-thinning procedures. The scaled age of a cluster is again of a random order of magnitude. Third, we construct a transformed Fisher--Wright tree, which (in the long-time limit) describes the structure of the space-time process associated with our system. All described phenomena are independent of the diffusion coefficient and occur for a large class of initial configurations (universality).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "interacting diffusion, clustering, infinite particle system, delayed coalescing random walk with immigration, transformed Fisher--Wright tree, low dimensional systems, ensemble of log-coalescents", } @Article{Bryc:1996:CMR, author = "W{\l}odzimierz Bryc", title = "Conditional Moment Representations for Dependent Random Variables", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "7:1--7:14", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-7", ISSN = "1083-6489", MRclass = "60A10 (60B99 60E15 62J12)", MRnumber = "1386299 (97j:60004)", MRreviewer = "M. M. Rao", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/7", abstract = "The question considered in this paper is which sequences of $p$-integrable random variables can be represented as conditional expectations of a fixed random variable with respect to a given sequence of sigma-fields. For finite families of sigma-fields, explicit inequality equivalent to solvability is stated; sufficient conditions are given for finite and infinite families of sigma-fields, and explicit expansions are presented.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "alternating conditional expectation, inverse problems, ACE", } @Article{Liao:1996:ASE, author = "Xiao Xin Liao and Xuerong Mao", title = "Almost Sure Exponential Stability of Neutral Differential Difference Equations with Damped Stochastic Perturbations", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "8:1--8:16", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-8", ISSN = "1083-6489", MRclass = "60H10 (34K40)", MRnumber = "1386300 (97d:60100)", MRreviewer = "Tom{\'a}s Caraballo", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/8", abstract = "In this paper we shall discuss the almost sure exponential stability for a neutral differential difference equation with damped stochastic perturbations of the form $ d[x(t) - G(x(t - \tau))] = f(t, x(t), x(t - \tau))d t + \sigma (t) d w(t) $. Several interesting examples are also given for illustration. It should be pointed out that our results are even new in the case when $ \sigma (t) \equiv 0 $, i.e., for deterministic neutral differential difference equations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "neutral equations, stochastic perturbation, exponential martingale inequality, Borel--Cantelli's lemma, Lyapunov exponent", } @Article{Roberts:1996:QBC, author = "Gareth O. Roberts and Jeffrey S. Rosenthal", title = "Quantitative bounds for convergence rates of continuous time {Markov} processes", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "9:1--9:21", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-9", ISSN = "1083-6489", MRclass = "60J25", MRnumber = "1423462 (97k:60198)", MRreviewer = "Mu Fa Chen", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/9", abstract = "We develop quantitative bounds on rates of convergence for continuous-time Markov processes on general state spaces. Our methods involve coupling and shift-coupling, and make use of minorization and drift conditions. In particular, we use auxiliary coupling to establish the existence of small (or pseudo-small) sets. We apply our method to some diffusion examples. We are motivated by interest in the use of Langevin diffusions for Monte Carlo simulation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov process, rates of convergence, coupling, shift-coupling, minorization condition, drift condition", } @Article{Arous:1996:MTD, author = "G{\'e}rard Ben Arous and Rapha{\"e}l Cerf", title = "Metastability of the Three Dimensional {Ising} Model on a Torus at Very Low Temperatures", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "10:1--10:55", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-10", ISSN = "1083-6489", MRclass = "82C44 (05B50 60J10 60K35)", MRnumber = "1423463 (98a:82086)", MRreviewer = "Peter Eichelsbacher", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/10; http://www.math.washington.edu/~ejpecp/EjpVol1/paper10.abs.html", abstract = "We study the metastability of the stochastic three dimensional Ising model on a finite torus under a small positive magnetic field at very low temperatures.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Ising, metastability, droplet, Freidlin--Wentzell theory, large deviations", } @Article{Bass:1996:USE, author = "Richard F. Bass", title = "Uniqueness for the {Skorokhod} equation with normal reflection in {Lipschitz} domains", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "11:1--11:29", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-11", ISSN = "1083-6489", MRclass = "60J60 (60J50)", MRnumber = "1423464 (98d:60155)", MRreviewer = "Zhen-Qing Chen", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/11; http://www.math.washington.edu/~ejpecp/EjpVol1/paper11.abs.html", abstract = "We consider the Skorokhod equation\par $$ d X_t = d W_t + (1 / 2) \nu (X_t), d L_t $$ in a domain $D$, where $ W_t$ is Brownian motion in $ R^d$, $ \nu $ is the inward pointing normal vector on the boundary of $D$, and $ L_t$ is the local time on the boundary. The solution to this equation is reflecting Brownian motion in $D$. In this paper we show that in Lipschitz domains the solution to the Skorokhod equation is unique in law.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Lipschitz domains, Neumann problem, reflecting Brownian motion, mixed boundary problem, Skorokhod equation, weak uniqueness, uniqueness in law, submartingale problem", } @Article{Gravner:1996:PTT, author = "Janko Gravner", title = "Percolation Times in Two-Dimensional Models For Excitable Media", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "12:1--12:19", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-12", ISSN = "1083-6489", MRclass = "60K35 (90C27)", MRnumber = "1423465 (98c:60141)", MRreviewer = "Rahul Roy", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/12", abstract = "The three-color {\em Greenberg--Hastings model (GHM) } is a simple cellular automaton model for an excitable medium. Each site on the lattice $ Z^2 $ is initially assigned one of the states 0, 1 or 2. At each tick of a discrete--time clock, the configuration changes according to the following synchronous rule: changes $ 1 \to 2 $ and $ 2 \to 0 $ are automatic, while an $x$ in state 0 may either stay in the same state or change to 1, the latter possibility occurring iff there is at least one representative of state 1 in the local neighborhood of $x$. Starting from a product measure with just 1's and 0's such dynamics quickly die out (turn into 0's), but not before 1's manage to form infinite connected sets. A very precise description of this ``transient percolation'' phenomenon can be obtained when the neighborhood of $x$ consists of 8 nearest points, the case first investigated by S. Fraser and R. Kapral. In addition, first percolation times for related monotone models are addressed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "additive growth dynamics, excitable media, Greenberg--Hastings model, percolation", } @Article{Lawler:1996:CTS, author = "Gregory F. Lawler", title = "Cut Times for Simple Random Walk", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "13:1--13:24", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-13", ISSN = "1083-6489", MRclass = "60J15 (60J65)", MRnumber = "1423466 (97i:60088)", MRreviewer = "Thomas Polaski", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/13", abstract = "Let $ S(n) $ be a simple random walk taking values in $ Z^d $. A time $n$ is called a cut time if \par $$ S[0, n] \cap S[n + 1, \infty) = \emptyset . $$ We show that in three dimensions the number of cut times less than $n$ grows like $ n^{1 - \zeta }$ where $ \zeta = \zeta_d$ is the intersection exponent. As part of the proof we show that in two or three dimensions \par $$ P(S[0, n] \cap S[n + 1, 2 n] = \emptyset) \sim n^{- \zeta }, $$ where $ \sim $ denotes that each side is bounded by a constant times the other side.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random walk, cut points, intersection exponent", } @Article{Dawson:1996:MST, author = "Donald A. Dawson and Andreas Greven", title = "Multiple Space-Time Scale Analysis For Interacting Branching Models", journal = j-ELECTRON-J-PROBAB, volume = "1", pages = "14:1--14:84", year = "1996", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v1-14", ISSN = "1083-6489", MRclass = "60K35 (60J80)", MRnumber = "1423467 (97m:60148)", MRreviewer = "Jean Vaillancourt", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/14", abstract = "We study a class of systems of countably many linearly interacting diffusions whose components take values in $ [0, \inf) $ and which in particular includes the case of interacting (via migration) systems of Feller's continuous state branching diffusions. The components are labelled by a hierarchical group. The longterm behaviour of this system is analysed by considering space-time renormalised systems in a combination of slow and fast time scales and in the limit as an interaction parameter goes to infinity. This leads to a new perspective on the large scale behaviour (in space and time) of critical branching systems in both the persistent and non-persistent cases and including that of the associated historical process. Furthermore we obtain an example for a rigorous renormalization analysis.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Branching processes, interacting diffusions, super random walk, renormalization, historical processes", } @Article{Takacs:1997:RWP, author = "Christiane Takacs", title = "Random Walk on Periodic Trees", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "1:1--1:16", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-15", ISSN = "1083-6489", MRclass = "60J15", MRnumber = "1436761 (97m:60101)", MRreviewer = "Jochen Geiger", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/15", abstract = "Following Lyons (1990, Random Walks and Percolation on Trees) we define a periodic tree, restate its branching number and consider a biased random walk on it. In the case of a transient walk, we describe the walk-invariant random periodic tree and calculate the asymptotic rate of escape (speed) of the walk. This is achieved by exploiting the connections between random walks and electric networks.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Trees, Random Walk, Speed", } @Article{Rosen:1997:LIL, author = "Jay Rosen", title = "Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "2:1--2:32", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-16", ISSN = "1083-6489", MRclass = "60F15 (60J15)", MRnumber = "1444245 (98d:60063)", MRreviewer = "Karl Grill", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/16", abstract = "Let $ X = X_n, X' = X'_n $, and $ X'' = X''_n $, $ n \geq 1 $, be three independent copies of a symmetric three dimensional random walk with $ E(|X_1 |^2 \log_+ |X_1 |) $ finite. In this paper we study the asymptotics of $ I_n $, the number of triple intersections up to step $n$ of the paths of $ X, X'$ and $ X''$ as $n$ goes to infinity. Our main result says that the limsup of $ I_n$ divided by $ \log (n) \log_3 (n)$ is equal to $ 1 \over \pi |Q|$, a.s., where $Q$ denotes the covariance matrix of $ X_1$. A similar result holds for $ J_n$, the number of points in the triple intersection of the ranges of $ X, X'$ and $ X''$ up to step $n$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random walks, intersections", } @Article{Abraham:1997:APB, author = "Romain Abraham and Wendelin Werner", title = "Avoiding-probabilities for {Brownian} snakes and super-{Brownian} motion", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "3:1--3:27", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-17", ISSN = "1083-6489", MRclass = "60J25 (60G57)", MRnumber = "1447333 (98j:60100)", MRreviewer = "John Verzani", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/17", abstract = "We investigate the asymptotic behaviour of the probability that a normalized $d$-dimensional Brownian snake (for instance when the life-time process is an excursion of height 1) avoids 0 when starting at distance $ \varepsilon $ from the origin. In particular we show that when $ \varepsilon $ tends to 0, this probability respectively behaves (up to multiplicative constants) like $ \varepsilon^4$, $ \varepsilon^{2 \sqrt {2}}$ and $ \varepsilon^{(\sqrt {17} - 1) / 2}$, when $ d = 1$, $ d = 2$ and $ d = 3$. Analogous results are derived for super-Brownian motion started from $ \delta_x$ (conditioned to survive until some time) when the modulus of $x$ tends to 0.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian snakes, superprocesses, non-linear differential equations", } @Article{Jakubowski:1997:NST, author = "Adam Jakubowski", title = "A non-{Skorohod} topology on the {Skorohod} space", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "4:1--4:21", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-18", ISSN = "1083-6489", MRclass = "60F17 (60B05 60B10 60G17)", MRnumber = "1475862 (98k:60046)", MRreviewer = "Ireneusz Szyszkowski", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/18", abstract = "A new topology (called $S$) is defined on the space $D$ of functions $ x \colon [0, 1] \to R^1$ which are right-continuous and admit limits from the left at each $ t > 0$. Although $S$ cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorohod's topologies $ J_1$ and $ M_1$. In particular, on the space $ P(D)$ of laws of stochastic processes with trajectories in $D$ the topology $S$ induces a sequential topology for which both the direct and the converse Prokhorov's theorems are valid, the a.s. Skorohod representation for subsequences exists and finite dimensional convergence outside a countable set holds.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Skorohod space, Skorohod representation, convergence in distribution, sequential spaces, semimartingales", } @Article{Arcones:1997:LIL, author = "Miguel A. Arcones", title = "The Law of the Iterated Logarithm for a Triangular Array of Empirical Processes", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "5:1--5:39", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-19", ISSN = "1083-6489", MRclass = "60B12 (60F15)", MRnumber = "1475863 (98k:60006)", MRreviewer = "Winfried Stute", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/19", abstract = "We study the compact law of the iterated logarithm for a certain type of triangular arrays of empirical processes, appearing in statistics (M-estimators, regression, density estimation, etc). We give necessary and sufficient conditions for the law of the iterated logarithm of these processes of the type of conditions used in Ledoux and Talagrand (1991): convergence in probability, tail conditions and total boundedness of the parameter space with respect to certain pseudometric. As an application, we consider the law of the iterated logarithm for a class of density estimators. We obtain the order of the optimal window for the law of the iterated logarithm of density estimators. We also consider the compact law of the iterated logarithm for kernel density estimators when they have large deviations similar to those of a Poisson process.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Empirical process, law of the iterated logarithm, triangular array, density estimation", } @Article{Bertoin:1997:CPV, author = "Jean Bertoin", title = "{Cauchy}'s Principal Value of Local Times of {L{\'e}vy} Processes with no Negative Jumps via Continuous Branching Processes", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "6:1--6:12", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-20", ISSN = "1083-6489", MRclass = "60J30 (60J55)", MRnumber = "1475864 (99b:60120)", MRreviewer = "N. H. Bingham", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/20", abstract = "Let $X$ be a recurrent L{\'e}vy process with no negative jumps and $n$ the measure of its excursions away from $0$. Using Lamperti's connection that links $X$ to a continuous state branching process, we determine the joint distribution under $n$ of the variables $ C^+_T = \int_0^T{\bf 1}_{{X_s > 0}}X_s^{-1}d s$ and $ C^-_T = \int_0^T{\bf 1}_{{X_s < 0}}|X_s|^{-1}d s$, where $T$ denotes the duration of the excursion. This provides a new insight on an identity of Fitzsimmons and Getoor on the Hilbert transform of the local times of $X$. Further results in the same vein are also discussed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Cauchy's principal value, L{\'e}vy process with no negative jumps, branching process", } @Article{Mueller:1997:FWR, author = "Carl Mueller and Roger Tribe", title = "Finite Width For a Random Stationary Interface", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "7:1--7:27", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-21", ISSN = "1083-6489", MRclass = "60H15 (35R60)", MRnumber = "1485116 (99g:60106)", MRreviewer = "Richard B. Sowers", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/21", abstract = "We study the asymptotic shape of the solution $ u(t, x) \in [0, 1] $ to a one-dimensional heat equation with a multiplicative white noise term. At time zero the solution is an interface, that is $ u(0, x) $ is 0 for all large positive $x$ and $ u(0, x)$ is 1 for all large negative $x$. The special form of the noise term preserves this property at all times $ t \geq 0$. The main result is that, in contrast to the deterministic heat equation, the width of the interface remains stochastically bounded.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic partial differential equations, duality, travelling waves, white noise", } @Article{Kager:1997:GOS, author = "Gerald Kager and Michael Scheutzow", title = "Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations", journal = j-ELECTRON-J-PROBAB, volume = "2", pages = "8:1--8:17", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v2-22", ISSN = "1083-6489", MRclass = "60H10 (28D10 34C35 34F05)", MRnumber = "1485117 (99b:60080)", MRreviewer = "Xue Rong Mao", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/22", abstract = "Let $Z$ be an $ R^m$-valued semimartingale with stationary increments which is realized as a helix over a filtered metric dynamical system $S$. Consider a stochastic differential equation with Lipschitz coefficients which is driven by $Z$. We show that its solution semiflow $ \phi $ has a version for which $ \varphi (t, \omega) = \phi (0, t, \omega)$ is a cocycle and therefore ($S$, $ \varphi $) is a random dynamical system. Our results generalize previous results which required $Z$ to be continuous. We also address the case of local Lipschitz coefficients with possible blow-up in finite time. Our abstract perfection theorems are designed to cover also potential applications to infinite dimensional equations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic differential equation, random dynamical system, cocycle, perfection", } @Article{Chaleyat-Maurel:1997:PPD, author = "Mireille Chaleyat-Maurel and David Nualart", title = "Points of Positive Density for Smooth Functionals", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "1:1--1:8", year = "1997", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-23", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/23", abstract = "In this paper we show that the set of points where the density of a Wiener functional is strictly positive is an open connected set, assuming some regularity conditions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Nondegenerate smooth Wiener functionals, Malliavin calculus, Support of the law", } @Article{Chaleyat-Maurel:1998:PPD, author = "Mireille Chaleyat-Maurel and David Nualart", title = "Points of positive density for smooth functionals", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "1:1--1:8", year = "1998", CODEN = "????", ISSN = "1083-6489", MRclass = "60H07", MRnumber = "1487202 (99b:60072)", MRreviewer = "Shi Zan Fang", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://www.math.washington.edu/~ejpecp/EjpVol3/paper1.abs.html", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Hitczenko:1998:HCM, author = "Pawe{\l} Hitczenko and Stanis{\l}aw Kwapie{\'n} and Wenbo V. Li and Gideon Schechtman and Thomas Schlumprecht and Joel Zinn", title = "Hypercontractivity and Comparison of Moments of Iterated Maxima and Minima of Independent Random Variables", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "2:1--2:26", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-24", ISSN = "1083-6489", MRclass = "60B11 (52A21 60E07 60E15 60G15)", MRnumber = "1491527 (99k:60008)", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/24", abstract = "We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "hypercontractivity, comparison of moments, iterated maxima and minima, Gaussian correlation inequalities, small ball probabilities", } @Article{Aldous:1998:EBM, author = "David Aldous and Vlada Limic", title = "The Entrance Boundary of the Multiplicative Coalescent", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "3:1--3:59", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-25", ISSN = "1083-6489", MRclass = "60J50 (60J75)", MRnumber = "1491528 (99d:60086)", MRreviewer = "M. G. Shur", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/25", abstract = "The multiplicative coalescent $ X(t) $ is a $ l^2$-valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a {\em standard} version of this process starting with infinitesimally small clusters at time $ - \infty $. In this paper, stochastic calculus techniques are used to describe all versions $ (X(t); - \infty < t < \infty)$ of the multiplicative coalescent. Roughly, an extreme version is specified by translation and scale parameters, and a vector $ c \in l^3$ of relative sizes of large clusters at time $ - \infty $. Such a version may be characterized in three ways: via its $ t \to - \infty $ behavior, via a representation of the marginal distribution $ X(t)$ in terms of excursion-lengths of a L{\'e}vy-type process, or via a weak limit of processes derived from the standard version via a ``coloring'' construction.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov process, entrance boundary, excursion, L{\'e}vy process, random graph, stochastic coalescent, weak convergence", } @Article{Cranston:1998:GEU, author = "Michael Cranston and Yves {Le Jan}", title = "Geometric Evolution Under Isotropic Stochastic Flow", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "4:1--4:36", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-26", ISSN = "1083-6489", MRclass = "60H10 (60J60)", MRnumber = "1610230 (99c:60115)", MRreviewer = "R{\'e}mi L{\'e}andre", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/26", abstract = "Consider an embedded hypersurface $M$ in $ R^3$. For $ \Phi_t$ a stochastic flow of differomorphisms on $ R^3$ and $ x \in M$, set $ x_t = \Phi_t (x)$ and $ M_t = \Phi_t (M)$. In this paper we will assume $ \Phi_t$ is an isotropic (to be defined below) measure preserving flow and give an explicit description by SDE's of the evolution of the Gauss and mean curvatures, of $ M_t$ at $ x_t$. If $ \lambda_1 (t)$ and $ \lambda_2 (t)$ are the principal curvatures of $ M_t$ at $ x_t$ then the vector of mean curvature and Gauss curvature, $ (\lambda_1 (t) + \lambda_2 (t)$, $ \lambda_1 (t) \lambda_2 (t))$, is a recurrent diffusion. Neither curvature by itself is a diffusion. In a separate addendum we treat the case of $M$ an embedded codimension one submanifold of $ R^n$. In this case, there are $ n - 1$ principal curvatures $ \lambda_1 (t), \ldots {}, \lambda_{n - 1} (t)$. If $ P_k, k = 1, \dots, n - 1$ are the elementary symmetric polynomials in $ \lambda_1, \ldots {}, \lambda_{n - 1}$, then the vector $ (P_1 (\lambda_1 (t), \ldots {}, \lambda_{n - 1} (t)), \ldots {}, P_{n - 1} (\lambda_1 (t), \ldots {}, \lambda_{n - 1} (t))$ is a diffusion and we compute the generator explicitly. Again no projection of this diffusion onto lower dimensions is a diffusion. Our geometric study of isotropic stochastic flows is a natural offshoot of earlier works by Baxendale and Harris (1986), LeJan (1985, 1991) and Harris (1981).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic flows, Lyapunov exponents, principal curvatures", } @Article{Evans:1998:CLT, author = "Steven N. Evans and Edwin A. Perkins", title = "Collision Local Times, Historical Stochastic Calculus, and Competing Species", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "5:1--5:120", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-27", ISSN = "1083-6489", MRclass = "60G57 (60H99 60J55 60J80)", MRnumber = "1615329 (99h:60098)", MRreviewer = "Anton Wakolbinger", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/27", abstract = "Branching measure-valued diffusion models are investigated that can be regarded as pairs of historical Brownian motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For 3 or fewer spatial dimensions, such processes are constructed using a new fixed-point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure-valued diffusions. This existence and uniqueness is used to establish well-posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in 4 or more dimensions models with the analogous definition do not exist.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "super-process, super-Brownian motion, interaction, local time, historical process, measure-valued Markov branching process, stochastic calculus, martingale measure, random measure", xxtitle = "Collision local times, historical stochastic calculus, and competing superprocesses", } @Article{Ferrari:1998:FSS, author = "P. A. Ferrari and L. R. G. Fontes", title = "Fluctuations of a Surface Submitted to a Random Average Process", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "6:1--6:34", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-28", ISSN = "1083-6489", MRclass = "60K35", MRnumber = "1624854 (99e:60214)", MRreviewer = "T. M. Liggett", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/28", abstract = "We consider a hypersurface of dimension $d$ imbedded in a $ d + 1$ dimensional space. For each $ x \in Z^d$, let $ \eta_t(x) \in R$ be the height of the surface at site $x$ at time $t$. At rate $1$ the $x$-th height is updated to a random convex combination of the heights of the `neighbors' of $x$. The distribution of the convex combination is translation invariant and does not depend on the heights. This motion, named the random average process (RAP), is one of the linear processes introduced by Liggett (1985). Special cases of RAP are a type of smoothing process (when the convex combination is deterministic) and the voter model (when the convex combination concentrates on one site chosen at random). We start the heights located on a hyperplane passing through the origin but different from the trivial one $ \eta (x) \equiv 0$. We show that, when the convex combination is neither deterministic nor concentrating on one site, the variance of the height at the origin at time $t$ is proportional to the number of returns to the origin of a symmetric random walk of dimension $d$. Under mild conditions on the distribution of the random convex combination, this gives variance of the order of $ t^{1 / 2}$ in dimension $ d = 1$, $ \log t$ in dimension $ d = 2$ and bounded in $t$ in dimensions $ d \ge 3$. We also show that for each initial hyperplane the process as seen from the height at the origin converges to an invariant measure on the hyper surfaces conserving the initial asymptotic slope. The height at the origin satisfies a central limit theorem. To obtain the results we use a corresponding probabilistic cellular automaton for which similar results are derived. This automaton corresponds to the product of (infinitely dimensional) independent random matrices whose rows are independent.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random average process, random surfaces, product of random matrices, linear process, voter model, smoothing process", } @Article{Feyel:1998:ASS, author = "Denis Feyel and Arnaud {de La Pradelle}", title = "On the approximate solutions of the {Stratonovitch} equation", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "7:1--7:14", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-29", ISSN = "1083-6489", MRclass = "60H07 (60G17)", MRnumber = "1624858 (99j:60075)", MRreviewer = "Marco Ferrante", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/29", abstract = "We present new methods for proving the convergence of the classical approximations of the Stratonovitch equation. We especially make use of the fractional Liouville-valued Sobolev space $ W^{r, p}({\cal J}_{\alpha, p}) $. We then obtain a support theorem for the capacity $ c_{r, p} $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stratonovitch equations, Kolmogorov lemma, quasi-sure analysis", } @Article{Capinski:1998:MAS, author = "Marek Capi{\'n}ski and Nigel J. Cutland", title = "Measure attractors for stochastic {Navier--Stokes} equations", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "8:1--8:15", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-30", ISSN = "1083-6489", MRclass = "60H15 (35B40 35Q30 35R60)", MRnumber = "1637081 (99f:60115)", MRreviewer = "Wilfried Grecksch", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/30", abstract = "We show existence of measure attractors for 2-D stochastic Navier--Stokes equations with general multiplicative noise.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic Navier--Stokes equations, measure attractors", } @Article{Kurtz:1998:MPC, author = "Thomas G. Kurtz", title = "Martingale problems for conditional distributions of {Markov} processes", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "9:1--9:29", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-31", ISSN = "1083-6489", MRclass = "60J25 (60G25 60G44 60J35)", MRnumber = "1637085 (99k:60186)", MRreviewer = "Amarjit Budhiraja", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/31", abstract = "Let $X$ be a Markov process with generator $A$ and let $ Y(t) = \gamma (X(t))$. The conditional distribution $ \pi_t$ of $ X(t)$ given $ \sigma (Y(s) \colon s \leq t)$ is characterized as a solution of a filtered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of Rogers and Pitman on Markov functions. Applications include uniqueness of filtering equations, exchangeability of the state distribution of vector-valued processes, verification of quasireversibility, and uniqueness for martingale problems for measure-valued processes. New results on the uniqueness of forward equations, needed in the proof of uniqueness for the filtered martingale problem are also presented.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "partial observation, conditional distribution, filtering, forward equation, martingale problem, Markov process, Markov function, quasireversibility, measure-valued process", } @Article{Kesten:1998:AAW, author = "Harry Kesten and Vladas Sidoravicius and Yu Zhang", title = "Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "10:1--10:75", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-32", ISSN = "1083-6489", MRclass = "60K35", MRnumber = "1637089 (99j:60155)", MRreviewer = "Rahul Roy", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/32", abstract = "We consider critical site percolation on the triangular lattice, that is, we choose $ X(v) = 0 $ or 1 with probability 1/2 each, independently for all vertices $v$ of the triangular lattice. We say that a word $ (\xi_1, \xi_2, \dots) \in \{ 0, 1 \}^{\mathbb {N}}$ is seen in the percolation configuration if there exists a selfavoiding path $ (v_1, v_2, \dots)$ on the triangular lattice with $ X(v_i) = \xi_i, i \ge 1$. We prove that with probability 1 ``almost all'' words, as well as all periodic words, except the two words $ (1, 1, 1, \dots)$ and $ (0, 0, 0, \dots)$, are seen. ``Almost all'' words here means almost all with respect to the measure $ \mu_\beta $ under which the $ \xi_i$ are i.i.d. with $ \mu_\beta {\xi_i = 0} = 1 - \mu_\beta {\xi_i = 1} = \beta $ (for an arbitrary $ 0 < \beta < 1$).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Percolation, Triangular lattice", } @Article{Yoo:1998:USS, author = "Hyek Yoo", title = "On the unique solvability of some nonlinear stochastic {PDEs}", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "11:1--11:22", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-33", ISSN = "1083-6489", MRclass = "60H15 (35R60)", MRnumber = "1639464 (99h:60126)", MRreviewer = "Bohdan Maslowski", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/33", abstract = "The Cauchy problem for 1-dimensional nonlinear stochastic partial differential equations is studied. The uniqueness and existence of solutions in $ c H^2_p(T)$-space are proved.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic PDEs, Space of Bessel potentials, Embedding theorems", } @Article{Fitzsimmons:1998:MPI, author = "P. J. Fitzsimmons", title = "{Markov} processes with identical bridges", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "12:1--12:12", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-34", ISSN = "1083-6489", MRclass = "60J25 (60J35)", MRnumber = "1641066 (99h:60142)", MRreviewer = "Kyle Siegrist", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/34", abstract = "Let $X$ and $Y$ be time-homogeneous Markov processes with common state space $E$, and assume that the transition kernels of $X$ and $Y$ admit densities with respect to suitable reference measures. We show that if there is a time $ t > 0$ such that, for each $ x \in E$, the conditional distribution of $ (X_s)_{0 \le s \leq t}$, given $ X_0 = x = X_t$, coincides with the conditional distribution of $ (Y_s)_{0 \leq s \leq t}$, given $ Y_0 = x = Y_t$, then the infinitesimal generators of $X$ and $Y$ are related by $ L^Y f = \psi^{-1}L^X(\psi f) - \lambda f$, where $ \psi $ is an eigenfunction of $ L^X$ with eigenvalue $ \lambda \in {\bf R}$. Under an additional continuity hypothesis, the same conclusion obtains assuming merely that $X$ and $Y$ share a ``bridge'' law for one triple $ (x, t, y)$. Our work extends and clarifies a recent result of I. Benjamini and S. Lee.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Bridge law, eigenfunction, transition density", } @Article{Davies:1998:LAE, author = "Ian M. Davies", title = "{Laplace} asymptotic expansions for {Gaussian} functional integrals", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "13:1--13:19", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-35", ISSN = "1083-6489", MRclass = "60H05 (41A60)", MRnumber = "1646472 (99i:60109)", MRreviewer = "Kun Soo Chang", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/35", abstract = "We obtain a Laplace asymptotic expansion, in orders of $ \lambda $, of\par $$ E^\rho_x \left \{ G(\lambda x) e^{- \lambda^{-2} F(\lambda x)} \right \} $$ the expectation being with respect to a Gaussian process. We extend a result of Pincus and build upon the previous work of Davies and Truman. Our methods differ from those of Ellis and Rosen in that we use the supremum norm to simplify the application of the result.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Gaussian processes, asymptotic expansions, functional integrals", } @Article{Csaki:1998:LFS, author = "Endre Cs{\'a}ki and Zhan Shi", title = "Large favourite sites of simple random walk and the {Wiener} process", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "14:1--14:31", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-36", ISSN = "1083-6489", MRclass = "60F15 (60G50 60J65)", MRnumber = "1646468 (2000d:60050)", MRreviewer = "Davar Khoshnevisan", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/36", abstract = "Let $ U(n) $ denote the most visited point by a simple symmetric random walk $ \{ S_k \}_{k \ge 0} $ in the first $n$ steps. It is known that $ U(n)$ and $ m a x_{0 \leq k \leq n} S_k$ satisfy the same law of the iterated logarithm, but have different upper functions (in the sense of P. L{\'e}vy). The distance between them however turns out to be transient. In this paper, we establish the exact rate of escape of this distance. The corresponding problem for the Wiener process is also studied.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Local time, favourite site, random walk, Wiener process", } @Article{Montgomery-Smith:1998:CRM, author = "Stephen Montgomery-Smith", title = "Concrete Representation of Martingales", journal = j-ELECTRON-J-PROBAB, volume = "3", pages = "15:1--15:15", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v3-37", ISSN = "1083-6489", MRclass = "60G42 (60G07 60H05)", MRnumber = "1658686 (99k:60116)", MRreviewer = "Dominique L{\'e}pingle", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/37", abstract = "Let $ (f_n) $ be a mean zero vector valued martingale sequence. Then there exist vector valued functions $ (d_n) $ from $ [0, 1]^n $ such that $ \int_0^1 d_n(x_1, \dots, x_n) \, d x_n = 0 $ for almost all $ x_1, \dots, x_{n - 1} $, and such that the law of $ (f_n) $ is the same as the law of $ (\sum_{k = 1}^n d_k(x_1, \dots, x_k)) $. Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "martingale, concrete representation, tangent sequence, condition (C.I.), UMD, Skorohod representation", } @Article{Pak:1998:RWF, author = "Igor Pak", title = "Random Walks On Finite Groups With Few Random Generators", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "1:1--1:11", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-38", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/38", abstract = "Let $G$ be a finite group. Choose a set $S$ of size $k$ uniformly from $G$ and consider a lazy random walk on the corresponding Cayley graph. We show that for almost all choices of $S$ given $ k = 2 a \, \log_2 |G|$, $ a > 1$, this walk mixes in under $ m = 2 a \, \log \frac {a}{a - 1} \log |G|$ steps. A similar result was obtained earlier by Alon and Roichman and also by Dou and Hildebrand using a different techniques. We also prove that when sets are of size $ k = \log_2 |G| + O(\log \log |G|)$, $ m = O(\log^3 |G|)$ steps suffice for mixing of the corresponding symmetric lazy random walk. Finally, when $G$ is abelian we obtain better bounds in both cases.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random random walks on groups, random subproducts, probabilistic method, separation distance", } @Article{Pak:1999:RWF, author = "Igor Pak", title = "Random walks on finite groups with few random generators", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "1:1--1:11", year = "1999", CODEN = "????", ISSN = "1083-6489", MRclass = "60B15 (60G50)", MRnumber = "1663526 (2000a:60008)", MRreviewer = "Martin V. Hildebrand", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://www.math.washington.edu/~ejpecp/EjpVol4/paper1.abs.html", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Krylov:1999:AVF, author = "N. V. Krylov", title = "Approximating Value Functions for Controlled Degenerate Diffusion Processes by Using Piece-Wise Constant Policies", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "2:1--2:19", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-39", ISSN = "1083-6489", MRclass = "49L25 (35K65)", MRnumber = "1668597 (2000b:49056)", MRreviewer = "Martino Bardi", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/39", abstract = "It is shown that value functions for controlled degenerate diffusion processes can be approximated with error of order $ h^{1 / 3} $ by using policies which are constant on intervals $ [k h^2, (k + 1)h^2) $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Bellman's equations, fully nonlinear equations", } @Article{Bressaud:1999:DCN, author = "Xavier Bressaud and Roberto Fern{\'a}ndez and Antonio Galves", title = "Decay of Correlations for Non-{H{\"o}lderian} Dynamics. {A} Coupling Approach", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "3:1--3:19", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-40", ISSN = "1083-6489", MRclass = "60G10 (28D05 37A25 37A50)", MRnumber = "1675304 (2000j:60049)", MRreviewer = "Bernard Schmitt", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/40", abstract = "We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non H{\"o}lder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Dynamical systems, non-H{\"o}lder dynamics, m ixing rate, chains with complete connections, relaxation speed, coupling methods", } @Article{Dawson:1999:HIF, author = "Donald A. Dawson and Andreas Greven", title = "Hierarchically interacting {Fleming--Viot} processes with selection and mutation: multiple space time scale analysis and quasi-equilibria", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "4:1--4:81", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-41", ISSN = "1083-6489", MRclass = "60J70 (60K35 92D10 92D25)", MRnumber = "1670873 (2000e:60139)", MRreviewer = "Anton Wakolbinger", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/41", abstract = "Genetic models incorporating resampling and migration are now fairly well-understood. Problems arise in the analysis, if both selection and mutation are incorporated. This paper addresses some aspects of this problem, in particular the analysis of the long-time behaviour before the equilibrium is reached (quasi-equilibrium, which is the time range of interest in most applications).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Interacting Fleming--Viot processes, Renormalization analysis, Selection, Mutation, Recombination", } @Article{Dohmen:1999:IIE, author = "Klaus Dohmen", title = "Improved Inclusion--Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "5:1--5:12", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-42", ISSN = "1083-6489", MRclass = "05A15 (05A19 05A20 68M15 90B25)", MRnumber = "1684161 (2000a:05009)", MRreviewer = "Stephen Tanny", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/42", abstract = "Recently, Naiman and Wynn introduced the concept of an abstract tube in order to obtain improved inclusion-exclusion identities and inequalities that involve much fewer terms than their classical counterparts. In this paper, we introduce a particular class of abstract tubes which plays an important role with respect to chromatic polynomials and network reliability. The inclusion-exclusion identities and inequalities associated with this class simultaneously generalize several well-known results such as Whitney's broken circuit theorem, Shier's expression for the reliability of a network as an alternating sum over chains in a semilattice and Narushima's inclusion-exclusion identity for posets. Moreover, we show that under some restrictive assumptions a polynomial time inclusion-exclusion algorithm can be devised, which generalizes an important result of Provan and Ball on network reliability.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Inclusion-exclusion, Bonferroni inequalities, sieve formula, abstract tube, abstract simplicial complex, partial order, chain, dynamic programming, graph coloring, chromatic polynomial, broken circuit complex, network reliability", } @Article{Dalang:1999:EMM, author = "Robert C. Dalang", title = "Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "6:1--6:29", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-43", ISSN = "1083-6489", MRclass = "60H05 (35R60 60G15 60G48 60H15)", MRnumber = "1684157 (2000b:60132)", MRreviewer = "Marta Sanz Sol{\'e}", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/43", abstract = "We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz distribution. This is the case for the wave equation in dimensions greater than two. Even when the integrand is a distribution, the value of our stochastic integral process is a real-valued martingale. We use this extended integral to recover necessary and sufficient conditions under which the linear wave equation driven by spatially homogeneous Gaussian noise has a process solution, and this in any spatial dimension. Under this condition, the non-linear three dimensional wave equation has a global solution. The same methods apply to the damped wave equation, to the heat equation and to various parabolic equations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic wave equation, stochastic heat equation, Gaussian noise, process solution", } @Article{Arcones:1999:WCR, author = "Miguel A. Arcones", title = "Weak Convergence for the Row Sums of a Triangular Array of Empirical Processes Indexed by a Manageable Triangular Array of Functions", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "7:1--7:17", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-44", ISSN = "1083-6489", MRclass = "60B12 (60F17)", MRnumber = "1684153 (2000c:60004)", MRreviewer = "Lajos Horv{\'a}th", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/44", abstract = "We study the weak convergence for the row sums of a general triangular array of empirical processes indexed by a manageable class of functions converging to an arbitrary limit. As particular cases, we consider random series processes and normalized sums of i.i.d. random processes with Gaussian and stable limits. An application to linear regression is presented. In this application, the limit of the row sum of a triangular array of empirical process is the mixture of a Gaussian process with a random series process.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Empirical processes, triangular arrays, manageable classes", } @Article{Worms:1999:MDS, author = "Julien Worms", title = "Moderate deviations for stable {Markov} chains and regression models", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "8:1--8:28", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-45", ISSN = "1083-6489", MRclass = "60F10 (60G10 62J02 62J05)", MRnumber = "1684149 (2000b:60073)", MRreviewer = "Peter Eichelsbacher", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/45", abstract = "We prove moderate deviations principles for \begin{itemize} \item unbounded additive functionals of the form $ S_n = \sum_{j = 1}^n g(X^{(p)}_{j - 1}) $, where $ (X_n)_{n \in N} $ is a stable $ R^d$-valued functional autoregressive model of order $p$ with white noise and stationary distribution $ \mu $, and $g$ is an $ R^q$-valued Lipschitz function of order $ (r, s)$; \item the error of the least squares estimator (LSE) of the matrix $ \theta $ in an $ R^d$-valued regression model $ X_n = \theta^t \phi_{n - 1} + \epsilon_n$, where $ (\epsilon_n)$ is a generalized Gaussian noise. \end{itemize} We apply these results to study the error of the LSE for a stable $ R^d$-valued linear autoregressive model of order $p$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Large and Moderate Deviations, Martingales, Markov Chains, Least Squares Estimator for a regression model", } @Article{Morters:1999:SSL, author = "Peter M{\"o}rters and Narn-Rueih Shieh", title = "Small scale limit theorems for the intersection local times of {Brownian} motion", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "9:1--9:23", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-46", ISSN = "1083-6489", MRclass = "60G17 (28A78 60J55 60J65)", MRnumber = "1690313 (2000e:60060)", MRreviewer = "Yimin Xiao", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/46", abstract = "In this paper we contribute to the investigation of the fractal nature of the intersection local time measure on the intersection of independent Brownian paths. We particularly point out the difference in the small scale behaviour of the intersection local times in three-dimensional space and in the plane by studying almost sure limit theorems motivated by the notion of average densities introduced by Bedford and Fisher. We show that in 3-space the intersection local time measure of two paths has an average density of order two with respect to the gauge function $ \varphi (r) = r $, but in the plane, for the intersection local time measure of p Brownian paths, the average density of order two fails to converge. The average density of order three, however, exists for the gauge function $ \varphi_p(r) = r^2 [\log (1 / r)]^p $. We also prove refined versions of the above results, which describe more precisely the fluctuations of the volume of small balls around these gauge functions by identifying the density distributions, or lacunarity distributions, of the intersection local times.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, intersection local time, Palm distribution, average density, density distribution, lacunarity distribution, logarithmic average", } @Article{Dembo:1999:TPT, author = "Amir Dembo and Yuval Peres and Jay Rosen and Ofer Zeitouni", title = "Thick Points for Transient Symmetric Stable Processes", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "10:1--10:13", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-47", ISSN = "1083-6489", MRclass = "60J55 (60G52)", MRnumber = "1690314 (2000f:60117)", MRreviewer = "Larbi Alili", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/47", abstract = "Let $ T(x, r) $ denote the total occupation measure of the ball of radius $r$ centered at $x$ for a transient symmetric stable processes of index $ b < d$ in $ R^d$ and $ K(b, d)$ denote the norm of the convolution with its 0-potential density, considered as an operator on $ L^2 (B(0, 1), d x)$. We prove that as $r$ approaches 0, almost surely $ \sup_{|x| \leq 1} T(x, r) / (r^b| \log r|) \to b K(b, d)$. Furthermore, for any $ a \in (0, b / K(b, d))$, the Hausdorff dimension of the set of ``thick points'' $x$ for which $ \limsup_{r \to 0} T(x, r) / (r^b | \log r|) = a$, is almost surely $ b - a / K(b, d)$; this is the correct scaling to obtain a nondegenerate ``multifractal spectrum'' for transient stable occupation measure. The liminf scaling of $ T(x, r)$ is quite different: we exhibit positive, finite, non-random $ c(b, d), C(b, d)$, such that almost surely $ c(b, d) < \sup_x \liminf_{r \to 0} T(x, r) / r^b < C(b, d)$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stable process, occupation measure, multifractal spectrum", } @Article{Pitman:1999:BMB, author = "Jim Pitman", title = "{Brownian} motion, bridge, excursion, and meander characterized by sampling at independent uniform times", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "11:1--11:33", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-48", ISSN = "1083-6489", MRclass = "60J65 (05A19 11B73)", MRnumber = "1690315 (2000e:60137)", MRreviewer = "G{\"o}tz Kersting", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/48; http://www.math.washington.edu/~ejpecp/EjpVol4/paper11.abs.html", abstract = "For a random process $X$ consider the random vector defined by the values of $X$ at times $ 0 < U_{n, 1} < \cdots {} < U_{n, n} < 1$ and the minimal values of $X$ on each of the intervals between consecutive pairs of these times, where the $ U_{n, i}$ are the order statistics of $n$ independent uniform $ (0, 1)$ variables, independent of $X$. The joint law of this random vector is explicitly described when $X$ is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at $n$ independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae are Brownian representations of various special functions, including Bessel polynomials, some hypergeometric polynomials, and the Hermite function. Various combinatorial identities involving random partitions and generalized Stirling numbers are also obtained.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "alternating exponential random walk, uniform order statistics, critical binary random tree, Vervaat's transformation, random partitions, generalized Stirling numbers, Bessel polynomials, McDonald function, products of gamma variables, Hermite function", } @Article{Greven:1999:LBB, author = "Andreas Greven and Achim Klenke and Anton Wakolbinger", title = "The Longtime Behavior of Branching Random Walk in a Catalytic Medium", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "12:1--12:80", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-49", ISSN = "1083-6489", MRclass = "60K35 (60J80)", MRnumber = "1690316 (2000a:60189)", MRreviewer = "T. M. Liggett", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/49", abstract = "Consider a countable collection of particles located on a countable group, performing a critical branching random walk where the branching rate of a particle is given by a random medium fluctuating both in space and time. Here we study the case where the time-space random medium (called catalyst) is also a critical branching random walk evolving autonomously while the local branching rate of the reactant process is proportional to the number of catalytic particles present at a site. The catalyst process and the reactant process typically have different underlying motions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Branching random walk in random medium, reactant-catalyst systems, interacting particle Systems, random media", } @Article{Peligrad:1999:CSS, author = "Magda Peligrad", title = "Convergence of Stopped Sums of Weakly Dependent Random Variables", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "13:1--13:13", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-50", ISSN = "1083-6489", MRclass = "60E15 (60F15 60G48)", MRnumber = "1692676 (2000d:60033)", MRreviewer = "Przemys{\l}aw Matu{\l}a", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/50", abstract = "In this paper we investigate stopped partial sums for weak dependent sequences.\par In particular, the results are used to obtain new maximal inequalities for strongly mixing sequences and related almost sure results.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Partial sums, maximal inequalities, weak dependent sequences, stopping times, amarts", } @Article{Steinsaltz:1999:RTC, author = "David Steinsaltz", title = "Random Time Changes for Sock-Sorting and Other Stochastic Process Limit Theorems", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "14:1--14:25", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-51", ISSN = "1083-6489", MRclass = "60F05 (60C05 60K05)", MRnumber = "1692672 (2000e:60038)", MRreviewer = "Lars Holst", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/51", abstract = "A common technique in the theory of stochastic process is to replace a discrete time coordinate by a continuous randomized time, defined by an independent Poisson or other process. Once the analysis is complete on this Poissonized process, translating the results back to the original setting may be nontrivial. It is shown here that, under fairly general conditions, if the process $ S_n $ and the time change $ \phi_n $ both converge, when normalized by the same constant, to limit processes combined process $ S_n(\phi_n(t)) $ converges, when properly normalized, to a sum of the limit of the original process, and the limit of the time change multiplied by the derivative of $ E S_n $. It is also shown that earlier results on the fine structure of the maxima are preserved by these time changes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "maximal inequalities, decoupling, Poissonization, functional central limit theorem, sorting, random allocations, auxiliary randomization, time change", } @Article{Pitman:1999:LMB, author = "Jim Pitman and Marc Yor", title = "The law of the maximum of a {Bessel} bridge", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "15:1--15:35", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-52", ISSN = "1083-6489", MRclass = "60J65 (33C10 60J60)", MRnumber = "1701890 (2000j:60101)", MRreviewer = "Endre Cs{\'a}ki", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/52; http://www.math.washington.edu/~ejpecp/EjpVol4/paper15.abs.html", abstract = "Let $ M_d $ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $ P(M_d \leq a)$ due to Gikhman and Kiefer for $ d = 1, 2, \ldots $ is shown to be valid for all real $ d > 0$. Various other characterizations of the distribution of $ M_d$ are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of $ M_d$ is described both as $d$ tends to infinity and as $d$ tends to zero.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian bridge, Brownian excursion, Brownian scaling, local time, Bessel process, zeros of Bessel functions, Riemann zeta function", } @Article{Igloi:1999:LRD, author = "E. Igl{\'o}i and G. Terdik", title = "Long-range dependence through gamma-mixed {Ornstein--Uhlenbeck} process", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "16:1--16:33", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-53", ISSN = "1083-6489", MRclass = "60H05 (60G15 60G18 60H10)", MRnumber = "1713649 (2000m:60060)", MRreviewer = "V. V. Anh", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/53", abstract = "The limit process of aggregational models---(i) sum of random coefficient AR(1) processes with independent Brownian motion (BM) inputs and (ii) sum of AR(1) processes with random coefficients of Gamma distribution and with input of common BM's, ---proves to be Gaussian and stationary and its transfer function is the mixture of transfer functions of Ornstein--Uhlenbeck (OU) processes by Gamma distribution. It is called Gamma-mixed Ornstein--Uhlenbeck process ($ \Gamma \mathsf {MOU}$). For independent Poisson alternating $0$-$1$ reward processes with proper random intensity it is shown that the standardized sum of the processes converges to the standardized $ \Gamma \mathsf {MOU}$ process. The $ \Gamma \mathsf {MOU}$ process has various interesting properties and it is a new candidate for the successful modelling of several Gaussian stationary data with long-range dependence. Possible applications and problems are also considered.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stationarity, Long-range dependence, Spectral representation, Ornstein--Uhlenbeck process, Aggregational model, Stochastic differentialequation, Fractional Brownian motion input, Heart rate variability", } @Article{Liptser:1999:MDT, author = "R. Liptser and V. Spokoiny", title = "Moderate Deviations Type Evaluation for Integral Functionals of Diffusion Processes", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "17:1--17:25", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-54", ISSN = "1083-6489", MRclass = "60F10 (60J60)", MRnumber = "1741723 (2001j:60054)", MRreviewer = "Anatolii A. Pukhal{\cprime}ski{\u\i}", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/54", abstract = "We establish a large deviations type evaluation for the family of integral functionals.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "large deviations, moderate deviations, diffusion", } @Article{Fukushima:1999:SMC, author = "Masatoshi Fukushima", title = "On semi-martingale characterizations of functionals of symmetric {Markov} processes", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "18:1--18:32", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-55", ISSN = "1083-6489", MRclass = "60J45 (31C25 60J55)", MRnumber = "1741537 (2001b:60091)", MRreviewer = "Zhen-Qing Chen", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/55", abstract = "For a quasi-regular (symmetric) Dirichlet space $ ({\cal E}, {\cal F}) $ and an associated symmetric standard process $ (X_t, P_x) $, we show that, for $ u i n {\cal F} $, the additive functional $ u^*(X_t) - u^*(X_0) $ is a semimartingale if and only if there exists an $ {\cal E}$-nest $ \{ F_n \} $ and positive constants $ C_n$ such that $ \vert {\cal E}(u, v) \vert \leq C_n \Vert v \Vert_\infty, v \in {\cal F}_{F_n, b}.$ In particular, a signed measure resulting from the inequality will be automatically smooth. One of the variants of this assertion is applied to the distorted Brownian motion on a closed subset of $ R^d$, giving stochastic characterizations of BV functions and Caccioppoli sets.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "quasi-regular Dirichlet form, strongly regular representation, additive functionals, semimartingale, smooth signed measure, BV function", } @Article{Getoor:1999:EGS, author = "Ronald K. Getoor", title = "An Extended Generator and {Schr{\"o}dinger} Equations", journal = j-ELECTRON-J-PROBAB, volume = "4", pages = "19:1--19:23", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v4-56", ISSN = "1083-6489", MRclass = "60J40 (60J25 60J35 60J45)", MRnumber = "1741538 (2001c:60115)", MRreviewer = "Zoran Vondra{\v{c}}ek", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/56", abstract = "The generator of a Borel right process is extended so that it maps functions to smooth measures. This extension may be defined either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schr{\"o}dinger equation with a (signed) measure serving as potential may be interpreted as an equation between measures. In this context general existence and uniqueness theorems for solutions are established. These are then specialized to obtain more concrete results in special situations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov processes, Schr{\"o}dinger equations, generators, smooth measures", } @Article{Sharpe:1999:MRS, author = "Michael Sharpe", title = "Martingales on Random Sets and the Strong Martingale Property", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "1:1--1:17", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-57", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/57", abstract = "Let $X$ be a process defined on an optional random set. The paper develops two different conditions on $X$ guaranteeing that it is the restriction of a uniformly integrable martingale. In each case, it is supposed that $X$ is the restriction of some special semimartingale $Z$ with canonical decomposition $ Z = M + A$. The first condition, which is both necessary and sufficient, is an absolute continuity condition on $A$. Under additional hypotheses, the existence of a martingale extension can be characterized by a strong martingale property of $X$. Uniqueness of the extension is also considered.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Martingale, random set, strong martingale property", } @Article{Camarri:1999:LDR, author = "Michael Camarri and Jim Pitman", title = "Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "2:1--2:18", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-58", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/58", abstract = "Given an arbitrary distribution on a countable set, consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent trials converge in distribution to an inhomogeneous continuum random tree.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Repeat times, point process, Poisson embedding, inhomogeneous continuum random tree, Rayleigh distribution", } @Article{Bessaih:1999:SWA, author = "Hakima Bessaih", title = "Stochastic Weak Attractor for a Dissipative {Euler} Equation", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "3:1--3:16", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-59", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/59", abstract = "In this paper a nonautonomous dynamical system is considered, a stochastic one that is obtained from the dissipative Euler equation subject to a stochastic perturbation, an additive noise. Absorbing sets have been defined as sets that depend on time and attracts from $ - \infty $. A stochastic weak attractor is constructed in phase space with respect to two metrics and is compact in the lower one.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Dissipative Euler Equation, random dynamical systems, attractors", } @Article{Bertoin:1999:TCD, author = "Jean Bertoin and Jim Pitman", title = "Two Coalescents Derived from the Ranges of Stable Subordinators", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "7:1--7:17", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-63", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/63", abstract = "Let $ M_\alpha $ be the closure of the range of a stable subordinator of index $ \alpha \in]0, 1 [ $. There are two natural constructions of the $ M_{\alpha } $'s simultaneously for all $ \alpha \in]0, 1 [ $, so that $ M_{\alpha } \subseteq M_{\beta } $ for $ 0 < \alpha < \beta < 1 $: one based on the intersection of independent regenerative sets and one based on Bochner's subordination. We compare the corresponding two coalescent processes defined by the lengths of complementary intervals of $ [0, 1] \backslash M_{1 - \rho } $ for $ 0 < \rho < 1 $. In particular, we identify the coalescent based on the subordination scheme with the coalescent recently introduced by Bolthausen and Sznitman.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "coalescent, stable, subordinator, Poisson--Dirichlet distribution", } @Article{Khoshnevisan:2000:LRF, author = "Davar Khoshnevisan and Yuval Peres and Yimin Xiao", title = "Limsup Random Fractals", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "4:1--4:24", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-60", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/60", abstract = "Orey and Taylor (1974) introduced sets of ``fast points'' where Brownian increments are exceptionally large, $ {\rm F}(\lambda) := \{ t \in [0, 1] \colon \limsup_{h \to 0}{ | X(t + h) - X(t)| / \sqrt { 2h| \log h|}} \ge \lambda \} $. They proved that for $ \lambda \in (0, 1] $, the Hausdorff dimension of $ {\rm F}(\lambda) $ is $ 1 - \lambda^2 $ a.s. We prove that for any analytic set $ E \subset [0, 1] $, the supremum of the $ \lambda $ such that $E$ intersects $ {\rm F}(\lambda)$ a.s. equals $ \sqrt {\text {dim}_p E }$, where $ \text {dim}_p E$ is the {\em packing dimension} of $E$. We derive this from a general result that applies to many other random fractals defined by limsup operations. This result also yields extensions of certain ``fractal functional limit laws'' due to Deheuvels and Mason (1994). In particular, we prove that for any absolutely continuous function $f$ such that $ f(0) = 0$ and the energy $ \int_0^1 |f'|^2 \, d t $ is lower than the packing dimension of $E$, there a.s. exists some $ t \in E$ so that $f$ can be uniformly approximated in $ [0, 1]$ by normalized Brownian increments $ s \mapsto [X(t + s h) - X(t)] / \sqrt { 2h| \log h|}$; such uniform approximation is a.s. impossible if the energy of $f$ is higher than the packing dimension of $E$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Limsup random fractal, packing dimension, Hausdorff dimension, Brownian motion, fast point", } @Article{Ichinose:2000:NED, author = "Takashi Ichinose and Satoshi Takanobu", title = "The Norm Estimate of the Difference Between the {Kac} Operator and {Schr{\"o}dinger} Semigroup {II}: The General Case Including the Relativistic Case", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "5:1--5:47", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-61", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/61", abstract = "More thorough results than in our previous paper in Nagoya Math. J. are given on the $ L_p$-operator norm estimates for the Kac operator $ e^{-tV / 2} e^{-tH_0} e^{-tV / 2}$ compared with the Schr{\"o}dinger semigroup $ e^{-t(H_0 + V)}$. The Schr{\"o}dinger operators $ H_0 + V$ to be treated in this paper are more general ones associated with the L{\'e}vy process, including the relativistic Schr{\"o}dinger operator. The method of proof is probabilistic based on the Feynman--Kac formula. It differs from our previous work in the point of using {\em the Feynman--Kac formula\/} not directly for these operators, but instead through {\em subordination\/} from the Brownian motion, which enables us to deal with all these operators in a unified way. As an application of such estimates the Trotter product formula in the $ L_p$-operator norm, with error bounds, for these Schr{\"o}dinger semigroups is also derived.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Schr{\"o}dinger operator, Schr{\"o}dinger semigroup, relativistic Schr{\"o}dinger operator, Trotter product formula, Lie--Trotter--Kato product formula, Feynman--Kac formula, subordination of Brownian motion, Kato's inequality", } @Article{Mikulevicius:2000:SEE, author = "R. Mikulevicius and G. Valiukevicius", title = "On Stochastic {Euler} equation in $ \mathbb {R}^d $", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "6:1--6:20", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-62", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/62", abstract = "Following the Arnold--Marsden--Ebin approach, we prove local (global in 2-D) existence and uniqueness of classical (H{\"o}lder class) solutions of stochastic Euler equation with random forcing.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic partial differential equations, Euler equation", } @Article{Lawler:2000:SCH, author = "Gregory Lawler", title = "Strict Concavity of the Half Plane Intersection Exponent for Planar {Brownian} Motion", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "8:1--8:33", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-64", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/64", abstract = "The intersection exponents for planar Brownian motion measure the exponential decay of probabilities of nonintersection of paths. We study the intersection exponent $ \xi (\lambda_1, \lambda_2) $ for Brownian motion restricted to a half plane which by conformal invariance is the same as Brownian motion restricted to an infinite strip. We show that $ \xi $ is a strictly concave function. This result is used in another paper to establish a universality result for conformally invariant intersection exponents.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, intersection exponent", } @Article{Conlon:2000:HEE, author = "Joseph Conlon and Ali Naddaf", title = "On Homogenization Of Elliptic Equations With Random Coefficients", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "9:1--9:58", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-65", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/65", abstract = "In this paper, we investigate the rate of convergence of the solution $ u_\varepsilon $ of the random elliptic partial difference equation $ (\nabla^{\varepsilon *} a(x / \varepsilon, \omega) \nabla^\varepsilon + 1)u_\varepsilon (x, \omega) = f(x) $ to the corresponding homogenized solution. Here $ x \in \varepsilon Z^d $, and $ \omega \in \Omega $ represents the randomness. Assuming that $ a(x) $'s are independent and uniformly elliptic, we shall obtain an upper bound $ \varepsilon^\alpha $ for the rate of convergence, where $ \alpha $ is a constant which depends on the dimension $ d \ge 2 $ and the deviation of $ a(x, \omega) $ from the identity matrix. We will also show that the (statistical) average of $ u_\varepsilon (x, \omega) $ and its derivatives decay exponentially for large $x$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Homogenization, elliptic equations, random environment, Euler-Lagrange equation", } @Article{Hu:2000:LCH, author = "Yueyun Hu", title = "The Laws of {Chung} and {Hirsch} for {Cauchy}'s Principal Values Related to {Brownian} Local Times", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "10:1--10:16", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-66", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/66", abstract = "Two Chung-type and Hirsch-type laws are established to describe the liminf asymptotic behaviours of the Cauchy's principal values related to Brownian local times. These results are generalized to a class of Brownian additive functionals.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Principal values, Brownian additive functional, liminf asymptotic behaviours", } @Article{Feyel:2000:ARP, author = "D. Feyel and A. {de La Pradelle}", title = "The Abstract {Riemannian} Path Space", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "11:1--11:17", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-67", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/67", abstract = "On the Wiener space $ \Omega $, we introduce an abstract Ricci process $ A_t $ and a pseudo-gradient $ F \rightarrow {F}^\sharp $ which are compatible through an integration by parts formula. They give rise to a $ \sharp $-Sobolev space on $ \Omega $, logarithmic Sobolev inequalities, and capacities, which are tight on Hoelder compact sets of $ \Omega $. These are then applied to the path space over a Riemannian manifold.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Wiener space, Sobolev spaces, Bismut--Driver formula, Logarithmic Sobolev inequality, Capacities, Riemannian manifold path space", } @Article{Schweinsberg:2000:CSM, author = "Jason Schweinsberg", title = "Coalescents with Simultaneous Multiple Collisions", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "12:1--12:50", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-68", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/68", abstract = "We study a family of coalescent processes that undergo ``simultaneous multiple collisions, '' meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. This family of processes, which we obtain from simple assumptions about the rates of different types of mergers, essentially coincides with a family of processes that Mohle and Sagitov obtain as a limit of scaled ancestral processes in a population model with exchangeable family sizes. We characterize the possible merger rates in terms of a single measure, show how these coalescents can be constructed from a Poisson process, and discuss some basic properties of these processes. This work generalizes some work of Pitman, who provides similar analysis for a family of coalescent processes in which many clusters can coalesce into a single cluster, but almost surely no two such mergers occur simultaneously.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "coalescence, ancestral processes, Poisson point processes, Markov processes, exchangeable random partitions", } @Article{Krylov:2000:SS, author = "N. Krylov", title = "{SPDEs} in {$ L_q((0, \tau], L_p) $} Spaces", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "13:1--13:29", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-69", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/69", abstract = "Existence and uniqueness theorems are presented for evolutional stochastic partial differential equations of second order in $ L_p$-spaces with weights allowing derivatives of solutions to blow up near the boundary. It is allowed for the powers of summability with respect to space and time variables to be different.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic partial differential equations, Sobolev spaces with weights", } @Article{Lyne:2000:TWC, author = "Owen Lyne", title = "Travelling Waves for a Certain First-Order Coupled {PDE} System", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "14:1--14:40", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-70", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/70", abstract = "This paper concentrates on a particular first-order coupled PDE system. It provides both a detailed treatment of the {\em existence\/} and {\em uniqueness\/} of monotone travelling waves to various equilibria, by differential-equation theory and by probability theory and a treatment of the corresponding hyperbolic initial-value problem, by analytic methods. The initial-value problem is studied using characteristics to show existence and uniqueness of a bounded solution for bounded initial data (subject to certain smoothness conditions). The concept of {\em weak\/} solutions to partial differential equations is used to rigorously examine bounded initial data with jump discontinuities. For the travelling wave problem the differential-equation treatment makes use of a shooting argument and explicit calculations of the eigenvectors of stability matrices. The probabilistic treatment is careful in its proofs of {\em martingale\/} (as opposed to merely local-martingale) properties. A modern {\em change-of-measure technique\/} is used to obtain the best lower bound on the speed of the monotone travelling wave --- with Heaviside initial conditions the solution converges to an approximate travelling wave of that speed (the solution tends to one ahead of the wave-front and to zero behind it). Waves to different equilibria are shown to be related by Doob $h$-transforms. {\em Large-deviation theory\/} provides heuristic links between alternative descriptions of minimum wave speeds, rigorous algebraic proofs of which are provided.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Travelling waves, Martingales, Branching processes", } @Article{Kopp:2000:CIM, author = "P. Kopp and Volker Wellmann", title = "Convergence in Incomplete Market Models", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "15:1--15:26", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-71", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/71", abstract = "The problem of pricing and hedging of contingent claims in incomplete markets has led to the development of various valuation methodologies. This paper examines the mean-variance approach to risk-minimisation and shows that it is robust under the convergence from discrete- to continuous-time market models. This property yields new convergence results for option prices, trading strategies and value processes in incomplete market models. Techniques from nonstandard analysis are used to develop new results for the lifting property of the minimal martingale density and risk-minimising strategies. These are applied to a number of incomplete market models:\par It is shown that the convergence of the underlying models implies the convergence of strategies and value processes for multinomial models and approximations of the Black--Scholes model by direct discretisation of the price process. The concept of $ D^2$-convergence is extended to these classes of models, including the construction of discretisation schemes. This yields new standard convergence results for these models.\par For ease of reference a summary of the main results from nonstandard analysis in the context of stochastic analysis is given as well as a brief introduction to mean-variance hedging and pricing.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Financial models, incomplete markets", } @Article{Goldsheid:2000:ECA, author = "Ilya Goldsheid and Boris Khoruzhenko", title = "Eigenvalue Curves of Asymmetric Tridiagonal Matrices", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "16:1--16:28", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-72", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/72", abstract = "Random Schr{\"o}dinger operators with imaginary vector potentials are studied in dimension one. These operators are non-Hermitian and their spectra lie in the complex plane. We consider the eigenvalue problem on finite intervals of length $n$ with periodic boundary conditions and describe the limit eigenvalue distribution when $n$ goes to infinity. We prove that this limit distribution is supported by curves in the complex plane. We also obtain equations for these curves and for the corresponding eigenvalue density in terms of the Lyapunov exponent and the integrated density of states of a ``reference'' symmetric eigenvalue problem. In contrast to these results, the spectrum of the limit operator in $ \ell^2 (Z)$ is a two dimensional set which is not approximated by the spectra of the finite-interval operators.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random matrix, Schr{\"o}dinger operator, Lyapunov exponent, eigenvalue distribution, complex eigenvalue.", } @Article{Geiger:2000:PPP, author = "Jochen Geiger", title = "{Poisson} point process limits in size-biased {Galton--Watson} trees", journal = j-ELECTRON-J-PROBAB, volume = "5", pages = "17:1--17:12", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v5-73", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/73", abstract = "Consider a critical binary continuous-time Galton--Watson tree size-biased according to the number of particles at time $t$. Decompose the population at $t$ according to the particles' degree of relationship with a distinguished particle picked purely at random from those alive at $t$. Keeping track of the times when the different families grow out of the distinguished line of descent and the related family sizes at $t$, we represent this relationship structure as a point process in a time-size plane. We study limits of these point processes in the single- and some multitype case.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Galton--Watson process, random tree, point process, limit laws", } @Article{Sengupta:2000:FPD, author = "Arindam Sengupta and Anish Sarkar", title = "Finitely Polynomially Determined {L{\'e}vy} Processes", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "7:1--7:22", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-80", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/80", abstract = "A time-space harmonic polynomial for a continuous-time process $ X = \{ X_t \colon t \ge 0 \} $ is a two-variable polynomial $P$ such that $ \{ P(t, X_t) \colon t \ge 0 \} $ is a martingale for the natural filtration of $X$. Motivated by L{\'e}vy's characterisation of Brownian motion and Watanabe's characterisation of the Poisson process, we look for classes of processes with reasonably general path properties in which a characterisation of those members whose laws are determined by a finite number of such polynomials is available. We exhibit two classes of processes, the first containing the L{\'e}vy processes, and the second a more general class of additive processes, with this property and describe the respective characterisations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "L{\'e}vy process, additive process, L{\'e}vy's characterisation, L{\'e}vy measure, Kolmogorov measure", } @Article{Mountford:2001:NLB, author = "Thomas Mountford", title = "A Note on Limiting Behaviour of Disastrous Environment Exponents", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "1:1--1:10", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-74", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/74", abstract = "We consider a random walk on the $d$-dimensional lattice and investigate the asymptotic probability of the walk avoiding a ``disaster'' (points put down according to a regular Poisson process on space-time). We show that, given the Poisson process points, almost surely, the chance of surviving to time $t$ is like $ e^{- \alpha \log (\frac 1k) t } $, as $t$ tends to infinity if $k$, the jump rate of the random walk, is small.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random walk, disaster point, Poisson process", } @Article{Su:2001:DCD, author = "Francis Su", title = "Discrepancy Convergence for the Drunkard's Walk on the Sphere", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "2:1--2:20", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-75", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/75", abstract = "We analyze the drunkard's walk on the unit sphere with step size $ \theta $ and show that the walk converges in order $ C / \sin^2 (\theta) $ steps in the discrepancy metric ($C$ a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "discrepancy, random walk, Gelfand pairs, homogeneous spaces, Legendre polynomials", } @Article{Bai:2001:LTN, author = "Zhi-Dong Bai and Hsien-Kuei Hwang and Wen-Qi Liang and Tsung-Hsi Tsai", title = "Limit Theorems for the Number of Maxima in Random Samples from Planar Regions", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "3:1--3:41", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-76", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/76", abstract = "We prove that the number of maximal points in a random sample taken uniformly and independently from a convex polygon is asymptotically normal in the sense of convergence in distribution. Many new results for other planar regions are also derived. In particular, precise Poisson approximation results are given for the number of maxima in regions bounded above by a nondecreasing curve.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Maximal points, multicriterial optimization, central limit theorems, Poisson approximations, convex polygons", } @Article{Kesten:2001:PAW, author = "Harry Kesten and Vladas Sidoravicius and Yu Zhang", title = "Percolation of Arbitrary words on the Close-Packed Graph of $ \mathbb {Z}^2 $", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "4:1--4:27", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-77", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/77", abstract = "Let $ {\mathbb {Z}}^2_{cp} $ be the close-packed graph of $ \mathbb {Z}^2 $, that is, the graph obtained by adding to each face of $ \mathbb {Z}^2 $ its diagonal edges. We consider site percolation on $ \mathbb {Z}^2_{cp} $, namely, for each $v$ we choose $ X(v) = 1$ or 0 with probability $p$ or $ 1 - p$, respectively, independently for all vertices $v$ of $ \mathbb {Z}^2_{cp}$. We say that a word $ (\xi_1, \xi_2, \dots) \in \{ 0, 1 \}^{\mathbb {N}}$ is seen in the percolation configuration if there exists a selfavoiding path $ (v_1, v_2, \dots)$ on $ \mathbb {Z}^2_{cp}$ with $ X(v_i) = \xi_i, i \ge 1$. $ p_c(\mathbb {Z}^2, \text {site})$ denotes the critical probability for site-percolation on $ \mathbb {Z}^2$. We prove that for each fixed $ p \in \big (1 - p_c(\mathbb {Z}^2, \text {site}), p_c(\mathbb {Z}^2, \text {site}) \big)$, with probability 1 all words are seen. We also show that for some constants $ C_i > 0$ there is a probability of at least $ C_1$ that all words of length $ C_0 n^2$ are seen along a path which starts at a neighbor of the origin and is contained in the square $ [ - n, n]^2$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Percolation, close-packing", } @Article{Flandoli:2001:SSS, author = "Franco Flandoli and Marco Romito", title = "Statistically Stationary Solutions to the {$3$D} {Navier--Stokes} Equations do not show Singularities", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "5:1--5:15", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-78", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/78", abstract = "If $ \mu $ is a probability measure on the set of suitable weak solutions of the 3D Navier--Stokes equations, invariant for the time-shift, with finite mean dissipation rate, then at every time $t$ the set of singular points is empty $ \mu $-a.s. The existence of a measure $ \mu $ with the previous properties is also proved; it may describe a turbulent asymptotic regime.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Navier--Stokes equations, suitable weak solutions, stationary solutions", } @Article{DeSantis:2001:SIP, author = "Emilio {De Santis}", title = "Strict Inequality for Phase Transition between Ferromagnetic and Frustrated Systems", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "6:1--6:27", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-79", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/79", abstract = "We consider deterministic and disordered frustrated systems in which we can show some strict inequalities with respect to related ferromagnetic systems. A case particularly interesting is the Edwards--Anderson spin-glass model in which it is possible to determine a region of uniqueness of the Gibbs measure, which is strictly larger than the region of uniqueness for the related ferromagnetic system. We analyze also deterministic systems with $ |J_b| \in [J_A, J_B] $ where $ 0 < J_A \leq J_B < \infty $, for which we prove strict inequality for the critical points of the related FK model. The results are obtained for the Ising models but some extensions to Potts models are possible.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Phase transition, Ising model, disordered systems, stochastic order", } @Article{Heck:2001:PLD, author = "Matthias Heck and Fa{\"\i}za Maaouia", title = "The Principle of Large Deviations for Martingale Additive Functionals of Recurrent {Markov} Processes", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "8:1--8:26", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-81", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/81", abstract = "We give a principle of large deviations for a generalized version of the strong central limit theorem. This generalized version deals with martingale additive functionals of a recurrent Markov process.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Central Limit Theorem (CLT), Large Deviations Principle (LDP), Markov Processes, Autoregressive Model (AR1), Positive Recurrent Processes, Martingale Additive Functional (MAF)", } @Article{Barlow:2001:TDA, author = "Martin Barlow and Takashi Kumagai", title = "Transition Density Asymptotics for Some Diffusion Processes with Multi-Fractal Structures", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "9:1--9:23", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-82", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/82", abstract = "We study the asymptotics as $ t \to 0 $ of the transition density of a class of $ \mu $-symmetric diffusions in the case when the measure $ \mu $ has a multi-fractal structure. These diffusions include singular time changes of Brownian motion on the unit cube.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Diffusion process, heat equation, transition density, spectral dimension, multi-fractal", } @Article{Pemantle:2001:WDB, author = "Robin Pemantle and Yuval Peres and Jim Pitman and Marc Yor", title = "Where Did the {Brownian} Particle Go?", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "10:1--10:22", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-83", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/83", abstract = "Consider the radial projection onto the unit sphere of the path a $d$-dimensional Brownian motion $W$, started at the center of the sphere and run for unit time. Given the occupation measure $ \mu $ of this projected path, what can be said about the terminal point $ W(1)$, or about the range of the original path? In any dimension, for each Borel set $A$ in $ S^{d - 1}$, the conditional probability that the projection of $ W(1)$ is in $A$ given $ \mu (A)$ is just $ \mu (A)$. Nevertheless, in dimension $ d \ge 3$, both the range and the terminal point of $W$ can be recovered with probability 1 from $ \mu $. In particular, for $ d \ge 3$ the conditional law of the projection of $ W(1)$ given $ \mu $ is not $ \mu $. In dimension 2 we conjecture that the projection of $ W(1)$ cannot be recovered almost surely from $ \mu $, and show that the conditional law of the projection of $ W(1)$ given $ \mu $ is not $ m u$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, conditional distribution of a path given its occupation measure, radial projection", } @Article{Fill:2001:MTM, author = "James Fill and Clyde {Schoolfield, Jr.}", title = "Mixing Times for {Markov} Chains on Wreath Products and Related Homogeneous Spaces", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "11:1--11:22", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-84", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/84", abstract = "We develop a method for analyzing the mixing times for a quite general class of Markov chains on the complete monomial group $ G \wr S_n $ and a quite general class of Markov chains on the homogeneous space $ (G \wr S_n) / (S_r \times S_{n - r}) $. We derive an exact formula for the $ L^2 $ distance in terms of the $ L^2 $ distances to uniformity for closely related random walks on the symmetric groups $ S_j $ for $ 1 \leq j \leq n $ or for closely related Markov chains on the homogeneous spaces $ S_{i + j} / (S_i \times S_j) $ for various values of $i$ and $j$, respectively. Our results are consistent with those previously known, but our method is considerably simpler and more general.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov chain, random walk, rate of convergence to stationarity, mixing time, wreath product, Bernoulli--Laplace diffusion, complete monomial group, hyperoctahedral group, homogeneous space, M{\"o}bius inversion.", } @Article{Mikulevicius:2001:NKT, author = "R. Mikulevicius and B. Rozovskii", title = "A Note on {Krylov}'s {$ L_p $}-Theory for Systems of SPDEs", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "12:1--12:35", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-85", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/85", abstract = "We extend Krylov's $ L_p$-solvability theory to the Cauchy problem for systems of parabolic stochastic partial differential equations. Some additional integrability and regularity properties are also presented.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic partial differential equations, Cauchy problem", } @Article{Nishioka:2001:BCO, author = "Kunio Nishioka", title = "Boundary Conditions for One-Dimensional Biharmonic Pseudo Process", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "13:1--13:27", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-86", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/86", abstract = "We study boundary conditions for a stochastic pseudo processes corresponding to the biharmonic operator. The biharmonic pseudo process ({\em BPP\/} for short). is composed, in a sense, of two different particles, a monopole and a dipole. We show how an initial-boundary problems for a 4-th order parabolic differential equation can be represented by {\em BPP\/} with various boundary conditions for the two particles: killing, reflection and stopping.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Boundary conditions for biharmonic pseudo process, killing, reflection, stopping", } @Article{Miermont:2001:OAC, author = "Gr{\'e}gory Miermont", title = "Ordered Additive Coalescent and Fragmentations Associated to {L{\'e}vy} Processes with No Positive Jumps", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "14:1--14:33", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-87", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/87", abstract = "We study here the fragmentation processes that can be derived from L{\'e}vy processes with no positive jumps in the same manner as in the case of a Brownian motion (cf. Bertoin [4]). One of our motivations is that such a representation of fragmentation processes by excursion-type functions induces a particular order on the fragments which is closely related to the additivity of the coalescent kernel. We identify the fragmentation processes obtained this way as a mixing of time-reversed extremal additive coalescents by analogy with the work of Aldous and Pitman [2], and we make its semigroup explicit.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Additive-coalescent, fragmentation, L{\'e}vy processes, processes with exchangeable increments", } @Article{Jonasson:2001:DPM, author = "Johan Jonasson", title = "On Disagreement Percolation and Maximality of the Critical Value for iid Percolation", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "15:1--15:13", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-88", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/88", abstract = "Two different problems are studied:\par \begin{itemize} \item For an infinite locally finite connected graph $G$, let $ p_c(G)$ be the critical value for the existence of an infinite cluster in iid bond percolation on $G$ and let $ P_c = \sup \{ p_c(G) \colon G \text { transitive }, p_c(G) < 1 \} $. Is $ P_c < 1$ ? \item Let $G$ be transitive with $ p_c(G) < 1$, take $ p \in [0, 1]$ and let $X$ and $Y$ be iid bond percolations on $G$ with retention parameters $ (1 + p) / 2$ and $ (1 - p) / 2$ respectively. Is there a $ q < 1$ such that $ p > q$ implies that for any monotone coupling $ (X', Y')$ of $X$ and $Y$ the edges for which $ X'$ and $ Y'$ disagree form infinite connected component(s) with positive probability? Let $ p_d(G)$ be the infimum of such $q$'s (including $ q = 1$) and let $ P_d = \sup \{ p_d(G) \colon G \text { transitive }, p_c(G) < 1 \} $. Is the stronger statement $ P_d < 1$ true? On the other hand: Is it always true that $ p_d(G) > p_c (G)$ ? \end{itemize}\par It is shown that if one restricts attention to biregular planar graphs then these two problems can be treated in a similar way and all the above questions are positively answered. We also give examples to show that if one drops the assumption of transitivity, then the answer to the above two questions is no. Furthermore it is shown that for any bounded-degree bipartite graph $G$ with $ p_c(G) < 1$ one has $ p_c(G) < p_d(G)$. Problem (2) arises naturally from [6] where an example is given of a coupling of the distinct plus- and minus measures for the Ising model on a quasi-transitive graph at super-critical inverse temperature. We give an example of such a coupling on the $r$-regular tree, $ {\bf T}_r$, for $ r > 1$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "coupling, Ising model, random-cluster model, transitive graph, planar graph", } @Article{DelMoral:2001:CDG, author = "P. {Del Moral} and M. Kouritzin and L. Miclo", title = "On a Class of Discrete Generation Interacting Particle Systems", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "16:1--16:26", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-89", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/89", abstract = "The asymptotic behavior of a general class of discrete generation interacting particle systems is discussed. We provide $ L_p$-mean error estimates for their empirical measure on path space and present sufficient conditions for uniform convergence of the particle density profiles with respect to the time parameter. Several examples including mean field particle models, genetic schemes and McKean's Maxwellian gases will also be given. In the context of Feynman--Kac type limiting distributions we also prove central limit theorems and we start a variance comparison for two generic particle approximating models.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Interacting particle systems, genetic algorithms, Feynman--Kac formulas, stochastic approximations, central limit theorem", } @Article{Kurtz:2001:SSF, author = "Thomas Kurtz and Richard Stockbridge", title = "Stationary Solutions and Forward Equations for Controlled and Singular Martingale Problems", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "17:1--17:52", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-90", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/90", abstract = "Stationary distributions of Markov processes can typically be characterized as probability measures that annihilate the generator in the sense that $ | \int_E A f d \mu = 0 $ for $ f \in {\cal D}(A) $; that is, for each such $ \mu $, there exists a stationary solution of the martingale problem for $A$ with marginal distribution $ \mu $. This result is extended to models corresponding to martingale problems that include absolutely continuous and singular (with respect to time) components and controls. Analogous results for the forward equation follow as a corollary.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "singular controls, stationary processes, Markov processes, martingale problems, forward equations, constrained Markov processes", } @Article{Atar:2001:IWT, author = "Rami Atar", title = "Invariant Wedges for a Two-Point Reflecting {Brownian} Motion and the ``Hot Spots'' Problem", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "18:1--18:19", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-91", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/91", abstract = "We consider domains $D$ of $ R^d$, $ d \ge 2$ with the property that there is a wedge $ V \subset R^d$ which is left invariant under all tangential projections at smooth portions of $ \partial D$. It is shown that the difference between two solutions of the Skorokhod equation in $D$ with normal reflection, driven by the same Brownian motion, remains in $V$ if it is initially in $V$. The heat equation on $D$ with Neumann boundary conditions is considered next. It is shown that the cone of elements $u$ of $ L^2 (D)$ satisfying $ u(x) - u(y) \ge 0$ whenever $ x - y \in V$ is left invariant by the corresponding heat semigroup. Positivity considerations identify an eigenfunction corresponding to the second Neumann eigenvalue as an element of this cone. For $ d = 2$ and under further assumptions, especially convexity of the domain, this eigenvalue is simple.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Reflecting Brownian motion, Neumann eigenvalue problem, convex domains", } @Article{Lambert:2001:JLA, author = "Amaury Lambert", title = "The Joint Law of Ages and Residual Lifetimes for Two Schemes of Regenerative Sets", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "19:1--19:23", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-92", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/92", abstract = "We are interested in the component intervals of the complements of a monotone sequence $ R_n \subseteq \dots \subseteq R_1 $ of regenerative sets, for two natural embeddings. One is based on Bochner's subordination, and one on the intersection of independent regenerative sets. For each scheme, we study the joint law of the so-called ages and residual lifetimes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Multivariate renewal theory, regenerative sets, subordinator, random covering intervals", } @Article{Lyne:2001:WSS, author = "Owen Lyne and David Williams", title = "Weak Solutions for a Simple Hyperbolic System", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "20:1--20:21", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-93", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/93", abstract = "The model studied concerns a simple first-order {\em hyperbolic\/} system. The solutions in which one is most interested have discontinuities which persist for all time, and therefore need to be interpreted as {\em weak\/} solutions. We demonstrate existence and uniqueness for such weak solutions, identifying a canonical `{\em exact\/}' solution which is {\em everywhere\/} defined. The direct method used is guided by the theory of measure-valued diffusions. The method is more effective than the method of characteristics, and has the advantage that it leads immediately to the McKean representation without recourse to It{\^o}'s formula. We then conduct computer studies of our model, both by integration schemes (which {\em do\/} use characteristics) and by `random simulation'.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Weak solutions, Travelling waves, Martingales, Branching processses", } @Article{Kolokoltsov:2001:SDF, author = "Vassili Kolokoltsov", title = "Small Diffusion and Fast Dying Out Asymptotics for Superprocesses as Non-{Hamiltonian} Quasiclassics for Evolution Equations", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "21:1--21:16", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-94", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/94", abstract = "The small diffusion and fast dying out asymptotics is calculated for nonlinear equations of a class of superprocesses on manifolds, and the corresponding logarithmic limit of the solution is shown to be given by a solution of a certain problem of calculus of variations with a non-additive (and non-integral) functional.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Dawson--Watanabe superprocess, reaction diffusion equation, logarithmic limit, small diffusion asymptotics, curvilinear Ornstein--Uhlenbeck process", } @Article{Telcs:2001:LSG, author = "Andras Telcs", title = "Local Sub-{Gaussian} Estimates on Graphs: The Strongly Recurrent Case", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "22:1--22:33", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-95", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/95", abstract = "This paper proves upper and lower off-diagonal, sub-Gaussian transition probabilities estimates for strongly recurrent random walks under sufficient and necessary conditions. Several equivalent conditions are given showing their particular role and influence on the connection between the sub-Gaussian estimates, parabolic and elliptic Harnack inequality.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random walks, potential theory, Harnack inequality, reversible Markov chains", } @Article{Benjamini:2001:RDL, author = "Itai Benjamini and Oded Schramm", title = "Recurrence of Distributional Limits of Finite Planar Graphs", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "23:1--23:13", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-96", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/96", abstract = "Suppose that $ G_j $ is a sequence of finite connected planar graphs, and in each $ G_j $ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $ G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random triangulations, random walks, mass transport, circle packing, volume growth", } @Article{Lototsky:2001:LSP, author = "Sergey Lototsky", title = "Linear Stochastic Parabolic Equations, Degenerating on the Boundary of a Domain", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "24:1--24:14", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-97", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/97", abstract = "A class of linear degenerate second-order parabolic equations is considered in arbitrary domains. It is shown that these equations are solvable using special weighted Sobolev spaces in essentially the same way as the non-degenerate equations in $ R^d $ are solved using the usual Sobolev spaces. The main advantages of this Sobolev-space approach are less restrictive conditions on the coefficients of the equation and near-optimal space-time regularity of the solution. Unlike previous works on degenerate equations, the results cover both classical and distribution solutions and allow the domain to be bounded or unbounded without any smoothness assumptions about the boundary. An application to nonlinear filtering of diffusion processes is discussed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "$L_p$ estimates, Weighted spaces, Nonlinear filtering", } @Article{Dawson:2001:SDS, author = "Donald Dawson and Zenghu Li and Hao Wang", title = "Superprocesses with Dependent Spatial Motion and General Branching Densities", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "25:1--25:33", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-98", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/98", abstract = "We construct a class of superprocesses by taking the high density limit of a sequence of interacting-branching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching density is given by an arbitrary bounded non-negative Borel function, and the superprocess is characterized by a martingale problem as a diffusion process with state space $ M({\bf R}) $, improving and extending considerably the construction of Wang (1997, 1998). It is then proved in a special case that a suitable rescaled process of the superprocess converges to the usual super Brownian motion. An extension to measure-valued branching catalysts is also discussed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "superprocess, interacting-branching particle system, diffusion process, martingale problem, dual process, rescaled limit, measure-valued catalyst", } @Article{Feyel:2001:FIF, author = "D. Feyel and A. {de La Pradelle}", title = "The {FBM} {It{\^o}}'s Formula Through Analytic Continuation", journal = j-ELECTRON-J-PROBAB, volume = "6", pages = "26:1--26:22", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v6-99", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/99", abstract = "The Fractional Brownian Motion can be extended to complex values of the parameter $ \alpha $ for $ \Re \alpha > {1 \over 2} $. This is a useful tool. Indeed, the obtained process depends holomorphically on the parameter, so that many formulas, as It{\^o} formula, can be extended by analytic continuation. For large values of $ \Re \alpha $, the stochastic calculus reduces to a deterministic one, so that formulas are very easy to prove. Hence they hold by analytic continuation for $ \Re \alpha \leq 1 $, containing the classical case $ \alpha = 1 $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Wiener space, Sobolev space, Stochastic integral, Fractional Brownian Motion, It{\^o}'s formula", } @Article{Jacka:2001:ECN, author = "Saul Jacka and Jon Warren", title = "Examples of Convergence and Non-convergence of {Markov} Chains Conditioned Not To Die", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "1:1--1:22", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-100", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/100", abstract = "In this paper we give two examples of evanescent Markov chains which exhibit unusual behaviour on conditioning to survive for large times. In the first example we show that the conditioned processes converge vaguely in the discrete topology to a limit with a finite lifetime, but converge weakly in the Martin topology to a non-Markovian limit. In the second example, although the family of conditioned laws are tight in the Martin topology, they possess multiple limit points so that weak convergence fails altogether.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Conditioned Markov process, evanescent process, Martin boundary, Martin topology, superharmonic function, Choquet representation, star, Kolmogorov K2 chain", } @Article{Lawler:2001:OAE, author = "Gregory Lawler and Oded Schramm and Wendelin Werner", title = "One-Arm Exponent for Critical {$2$D} Percolation", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "2:1--2:13", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-101", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/101", abstract = "The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R$ to the power $ 5 / 48$ as $R$ goes to infinity.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Percolation, critical exponents", } @Article{Darling:2001:ILP, author = "R. Darling", title = "Intrinsic Location Parameter of a Diffusion Process", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "3:1--3:23", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-102", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/102", abstract = "For nonlinear functions $f$ of a random vector $Y$, $ E[f(Y)]$ and $ f(E[Y])$ usually differ. Consequently the mathematical expectation of $Y$ is not intrinsic: when we change coordinate systems, it is not invariant. This article is about a fundamental and hitherto neglected property of random vectors of the form $ Y = f(X(t))$, where $ X(t)$ is the value at time $t$ of a diffusion process $X$: namely that there exists a measure of location, called the ``intrinsic location parameter'' (ILP), which coincides with mathematical expectation only in special cases, and which is invariant under change of coordinate systems. The construction uses martingales with respect to the intrinsic geometry of diffusion processes, and the heat flow of harmonic mappings. We compute formulas which could be useful to statisticians, engineers, and others who use diffusion process models; these have immediate application, discussed in a separate article, to the construction of an intrinsic nonlinear analog to the Kalman Filter. We present here a numerical simulation of a nonlinear SDE, showing how well the ILP formula tracks the mean of the SDE for a Euclidean geometry.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "intrinsic location parameter, gamma-martingale, stochastic differential equation, forward--backwards SDE, harmonic map, nonlinear heat equation", } @Article{Najim:2001:CTT, author = "Jamal Najim", title = "A {Cram{\'e}r} Type Theorem for Weighted Random Variables", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "4:1--4:32", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-103", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/103", abstract = "A Large Deviation Principle (LDP) is proved for the family $ (1 / n) \sum_1^n f(x_i^n) Z_i $ where $ (1 / n) \sum_1^n \delta_{x_i^n} $ converges weakly to a probability measure on $R$ and $ (Z_i)_{i \in N}$ are $ R^d$-valued independent and identically distributed random variables having some exponential moments, i.e.,\par $$ E e^{a |Z|} < \infty $$ for some $ 0 < a < \infty $. The main improvement of this work is the relaxation of the steepness assumption concerning the cumulant generating function of the variables $ (Z_i)_{i \in N}$. In fact, G{\"a}rtner-Ellis' theorem is no longer available in this situation. As an application, we derive a LDP for the family of empirical measures $ (1 / n) \sum_1^n Z_i \delta_{x_i^n}$. These measures are of interest in estimation theory (see Gamboa et al., Csiszar et al.), gas theory (see Ellis et al., van den Berg et al.), etc. We also derive LDPs for empirical processes in the spirit of Mogul'skii's theorem. Various examples illustrate the scope of our results.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Large Deviations, empirical means, empirical measures, maximum entropy on the means", } @Article{Konig:2001:NCR, author = "Wolfgang K{\"o}nig and Neil O'Connell and S{\'e}bastien Roch", title = "Non-Colliding Random Walks, Tandem Queues, and Discrete Orthogonal Polynomial Ensembles", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "5:1--5:24", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-104", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/104", abstract = "We show that the function $ h(x) = \prod_{i < j}(x_j - x_i) $ is harmonic for any random walk in $ R^k $ with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber $ W = \{ x \colon x_1 < x_2 < \cdots < x_k \} $ onto a point where $h$ vanishes, we define the corresponding Doob $h$-transform. For certain special cases, we show that the marginal distribution of the conditioned process at a fixed time is given by a familiar discrete orthogonal polynomial ensemble. These include the Krawtchouk and Charlier ensembles, where the underlying walks are binomial and Poisson, respectively. We refer to the corresponding conditioned processes in these cases as the Krawtchouk and Charlier processes. In [O'Connell and Yor (2001b)], a representation was obtained for the Charlier process by considering a sequence of $ M / M / 1$ queues in tandem. We present the analogue of this representation theorem for the Krawtchouk process, by considering a sequence of discrete-time $ M / M / 1$ queues in tandem. We also present related results for random walks on the circle, and relate a system of non-colliding walks in this case to the discrete analogue of the circular unitary ensemble (CUE).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Non-colliding random walks, tandem queues", } @Article{Zahle:2001:RBR, author = "Iljana Z{\"a}hle", title = "Renormalizations of Branching Random Walks in Equilibrium", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "7:1--7:57", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-106", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/106", abstract = "We study the $d$-dimensional branching random walk for $ d > 2$. This process has extremal equilibria for every intensity. We are interested in the large space scale and large space-time scale behavior of the equilibrium state. We show that the fluctuations of space and space-time averages with a non-classical scaling are Gaussian in the limit. For this purpose we use the historical process, which allows a family decomposition. To control the distribution of the families we use the concept of canonical measures and Palm distributions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Renormalization, branching random walk, Green's function of random walks, Palm distribution", } @Article{Luo:2001:STP, author = "S. Luo and John Walsh", title = "A Stochastic Two-Point Boundary Value Problem", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "12:1--12:32", year = "2001", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-111", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/111", abstract = "We investigate the two-point stochastic boundary-value problem on $ [0, 1] $: \begin{equation}\label{1} \begin{split} U'' &= f(U)\dot W + g(U, U')\\ U(0) &= \xi\\ U(1)&= \eta. \end{split} \tag{1} \end{equation} where $ \dot W $ is a white noise on $ [0, 1] $, $ \xi $ and $ \eta $ are random variables, and $f$ and $g$ are continuous real-valued functions. This is the stochastic analogue of the deterministic two point boundary-value problem, which is a classical example of bifurcation. We find that if $f$ and $g$ are affine, there is no bifurcation: for any r.v. $ \xi $ and $ \eta $, (1) has a unique solution a.s. However, as soon as $f$ is non-linear, bifurcation appears. We investigate the question of when there is either no solution whatsoever, a unique solution, or multiple solutions. We give examples to show that all these possibilities can arise. While our results involve conditions on $f$ and $g$, we conjecture that the only case in which there is no bifurcation is when $f$ is affine.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic boundary-value problems, bifurcations", } @Article{Diaconis:2002:RWT, author = "Persi Diaconis and Susan Holmes", title = "Random Walks on Trees and Matchings", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "6:1--6:17", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-105", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/105", abstract = "We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on $ 2 n $ vertices. Roughly, the results show that $ (1 / 2) n \log n $ steps are necessary and suffice to achieve randomness. The proof depends on the representation theory of the symmetric group and a bijection between trees and matchings.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov Chain, Matchings, Phylogenetic Tree, Fourier analysis, Zonal polynomials, Coagulation-Fragmentation", } @Article{Mayer-Wolf:2002:ACC, author = "Eddy Mayer-Wolf and Ofer Zeitouni and Martin Zerner", title = "Asymptotics of Certain Coagulation--Fragmentation Processes and Invariant {Poisson--Dirichlet} Measures", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "8:1--8:25", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-107", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/107", abstract = "We consider Markov chains on the space of (countable) partitions of the interval $ [0, 1] $, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $ \beta_m $ (if the sampled parts are distinct) or splitting the part with probability $ \beta_s $, according to a law $ \sigma $ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $ \sigma $ is the uniform measure, then the Poisson--Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of ``analytic'' invariant measures. We also derive transience and recurrence criteria for these chains.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Partitions, coagulation, fragmentation, invariant measures, Poisson--Dirichlet", } @Article{Evans:2002:ERW, author = "Steven Evans", title = "Eigenvalues of Random Wreath Products", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "9:1--9:15", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-108", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/108", abstract = "Consider a uniformly chosen element $ X_n $ of the $n$-fold wreath product $ \Gamma_n = G \wr G \wr \cdots \wr G$, where $G$ is a finite permutation group acting transitively on some set of size $s$. The eigenvalues of $ X_n$ in the natural $ s^n$-dimensional permutation representation (the composition representation) are investigated by considering the random measure $ \Xi_n$ on the unit circle that assigns mass $1$ to each eigenvalue. It is shown that if $f$ is a trigonometric polynomial, then $ \lim_{n \rightarrow \infty } P \{ \int f d \Xi_n \ne s^n \int f d \lambda \} = 0$, where $ \lambda $ is normalised Lebesgue measure on the unit circle. In particular, $ s^{-n} \Xi_n$ converges weakly in probability to $ \lambda $ as $ n \rightarrow \infty $. For a large class of test functions $f$ with non-terminating Fourier expansions, it is shown that there exists a constant $c$ and a non-zero random variable $W$ (both depending on $f$) such that $ c^{-n} \int f d \Xi_n$ converges in distribution as $ n \rightarrow \infty $ to $W$. These results have applications to Sylow $p$-groups of symmetric groups and autmorphism groups of regular rooted trees.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random permutation, random matrix, Haar measure, regular tree, Sylow, branching process, multiplicative function", } @Article{Mueller:2002:HPR, author = "Carl Mueller and Roger Tribe", title = "Hitting Properties of a Random String", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "10:1--10:29", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-109", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/109", abstract = "We consider Funaki's model of a random string taking values in $ \mathbf {R}^d $. It is specified by the following stochastic PDE,\par $$ \frac {\partial u(x)}{\partial t} = \frac {\partial^2 u(x)}{\partial x^2} + \dot {W}. $$ where $ \dot {W} = \dot {W}(x, t) $ is two-parameter white noise, also taking values in $ \mathbf {R}^d $. We find the dimensions in which the string hits points, and in which it has double points of various types. We also study the question of recurrence and transience.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Martingale, random set, strong martingale property", } @Article{Belitsky:2002:DSS, author = "Vladimir Belitsky and Gunter Sch{\"u}tz", title = "Diffusion and Scattering of Shocks in the Partially Asymmetric Simple Exclusion Process", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "11:1--11:21", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-110", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/110", abstract = "We study the behavior of shocks in the asymmetric simple exclusion process on $Z$ whose initial distribution is a product measure with a finite number of shocks. We prove that if the particle hopping rates of this process are in a particular relation with the densities of the initial measure then the distribution of this process at any time is a linear combination of shock measures of the structure similar to that of the initial distribution. The structure of this linear combination allows us to interpret this result by saying that the shocks of the initial distribution perform continuous time random walks on $Z$ interacting by the exclusion rule. We give explicit expressions for the hopping rates of these random walks. The result is derived with a help of quantum algebra technique. We made the presentation self-contained for the benefit of readers not acquainted with this approach, but interested in applying it in the study of interacting particle systems.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Asymmetric simple exclusion process, evolution of shock measures, quantum algebra", } @Article{Winter:2002:MSA, author = "Anita Winter", title = "Multiple Scale Analysis of Spatial Branching Processes under the Palm Distribution", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "13:1--13:74", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-112", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/112", abstract = "We consider two types of measure-valued branching processes on the lattice $ Z^d $. These are on the one hand side a particle system, called branching random walk, and on the other hand its continuous mass analogue, a system of interacting diffusions also called super random walk. It is known that the long-term behavior differs sharply in low and high dimensions: if $ d \leq 2 $ one gets local extinction, while, for $ d \geq 3 $, the systems tend to a non-trivial equilibrium. Due to Kallenberg's criterion, local extinction goes along with clumping around a 'typical surviving particle.' This phenomenon is called clustering. A detailed description of the clusters has been given for the corresponding processes on $ R^2 $ in Klenke (1997). Klenke proved that with the right scaling the mean number of particles over certain blocks are asymptotically jointly distributed like marginals of a system of coupled Feller diffusions, called system of tree indexed Feller diffusions, provided that the initial intensity is appropriately increased to counteract the local extinction. The present paper takes different remedy against the local extinction allowing also for state-dependent branching mechanisms. Instead of increasing the initial intensity, the systems are described under the Palm distribution. It will turn out together with the results in Klenke (1997) that the change to the Palm measure and the multiple scale analysis commute, as $ t \to \infty $. The method of proof is based on the fact that the tree indexed systems of the branching processes and of the diffusions in the limit are completely characterized by all their moments. We develop a machinery to describe the space-time moments of the superprocess effectively and explicitly.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "infinite particle system, superprocess, interacting diffusion, clustering, Palm distribution, grove indexed systems of diffusions, grove indexed systems of branching models, Kallenberg's backward tree", } @Article{Matsumoto:2002:WFS, author = "Hiroyuki Matsumoto and Setsuo Taniguchi", title = "{Wiener} Functionals of Second Order and Their {L{\'e}vy} Measures", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "14:1--14:30", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-113", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/113", abstract = "The distributions of Wiener functionals of second order are infinitely divisible. An explicit expression of the associated L{\'e}vy measures in terms of the eigenvalues of the corresponding Hilbert--Schmidt operators on the Cameron--Martin subspace is presented. In some special cases, a formula for the densities of the distributions is given. As an application of the explicit expression, an exponential decay property of the characteristic functions of the Wiener functionals is discussed. In three typical examples, complete descriptions are given.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Wiener functional of second order, L{\'e}vy measure, Mellin transform, exponential decay", } @Article{Dawson:2002:MCB, author = "Donald Dawson and Alison Etheridge and Klaus Fleischmann and Leonid Mytnik and Edwin Perkins and Jie Xiong", title = "Mutually Catalytic Branching in The Plane: Infinite Measure States", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "15:1--15:61", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-114", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/114", abstract = "A two-type infinite-measure-valued population in $ R^2 $ is constructed which undergoes diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a collision rate sufficiently small compared with the diffusion rate, the model is constructed as a pair of infinite-measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit (in law), local extinction of one type is shown. Moreover the surviving population is uniform with random intensity. The process constructed is a rescaled limit of the corresponding $ Z^2$-lattice model studied by Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior of that model under critical scaling. This part of a trilogy extends results from the finite-measure-valued case, whereas uniqueness questions are again deferred to the third part.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Catalyst, reactant, measure-valued branching, interactive branching, state-dependent branching, two-dimensional process, absolute continuity, self-similarity, collision measure, collision local time, martingale problem, moment equations, segregation of ty", } @Article{Alves:2002:PTF, author = "Oswaldo Alves and Fabio Machado and Serguei Popov", title = "Phase Transition for the Frog Model", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "16:1--16:21", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-115", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/115", abstract = "We study a system of simple random walks on graphs, known as {\em frog model}. This model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability $ 1 - p $. When an active particle hits a sleeping particle, the latter becomes active. Phase transition results and asymptotic values for critical parameters are presented for $ Z^d $ and regular trees.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "simple random walk, critical probability, percolation", } @Article{Abraham:2002:PSF, author = "Romain Abraham and Laurent Serlet", title = "{Poisson} Snake and Fragmentation", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "17:1--17:15", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-116", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/116", abstract = "Our main object that we call the Poisson snake is a Brownian snake as introduced by Le Gall. This process has values which are trajectories of standard Poisson process stopped at some random finite lifetime with Brownian evolution. We use this Poisson snake to construct a self-similar fragmentation as introduced by Bertoin. A similar representation was given by Aldous and Pitman using the Continuum Random Tree. Whereas their proofs used approximation by discrete models, our representation allows continuous time arguments.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Path-valued process, Brownian snake, Poisson process, fragmentation, coalescence, self-similarity", } @Article{Lejay:2002:CSI, author = "Antoine Lejay", title = "On the Convergence of Stochastic Integrals Driven by Processes Converging on account of a Homogenization Property", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "18:1--18:18", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-117", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/117", abstract = "We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Levy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of its terms. In fact, the answer may be negative, especially in presence of a highly oscillating first-order differential term. This provides us some counterexamples to the theory of good sequence of semimartingales.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic differential equations, good sequence of semimartingales, conditions UT and UCV, L{\'e}vy area", } @Article{Kolokoltsov:2002:TNE, author = "Vassili Kolokoltsov and R. L. Schilling and A. Tyukov", title = "Transience and Non-explosion of Certain Stochastic {Newtonian} Systems", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "19:1--19:19", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-118", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/118", abstract = "We give sufficient conditions for non-explosion and transience of the solution $ (x_t, p_t) $ (in dimensions $ \geq 3$) to a stochastic Newtonian system of the form\par $$ \begin {cases} d x_t = p_t \, d t, \\ d p_t = - \frac {\partial V(x_t) }{\partial x} \, d t - \frac { \partial c(x_t) }{ \partial x} \, d \xi_t, \end {cases} $$ where $ \{ \xi_t \}_{t \geq 0}$ is a $d$-dimensional L{\'e}vy process, $ d \xi_t$ is an It{\^o} differential and $ c \in C^2 (\mathbb {R}^d, \mathbb {R}^d)$, $ V \in C^2 (\mathbb {R}^d, \mathbb {R})$ such that $ V \geq 0$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "alpha-stable Levy processes; Levy processes; Non-explosion.; Stochastic Newtonian systems; Transience", } @Article{Fannjiang:2002:DLR, author = "Albert Fannjiang and Tomasz Komorowski", title = "Diffusion in Long-Range Correlated {Ornstein--Uhlenbeck} Flows", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "20:1--20:22", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-119", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/119", abstract = "We study a diffusion process with a molecular diffusion and random Markovian--Gaussian drift for which the usual (spatial) Peclet number is infinite. We introduce a temporal Peclet number and we prove that, under the finiteness of the temporal Peclet number, the laws of diffusions under the diffusive rescaling converge weakly, to the law of a Brownian motion. We also show that the effective diffusivity has a finite, nonzero limit as the molecular diffusion tends to zero.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Ornstein--Uhlenbeck flow, martingale central limit theorem, homogenization, Peclet number", } @Article{Warren:2002:NMP, author = "Jon Warren", title = "The Noise Made by a {Poisson} Snake", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "21:1--21:21", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-120", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/120", abstract = "The purpose of this article is to study a coalescing flow of sticky Brownian motions. Sticky Brownian motion arises as a weak solution of a stochastic differential equation, and the study of the flow reveals the nature of the extra randomness that must be added to the driving Brownian motion. This can be represented in terms of Poissonian marking of the trees associated with the excursions of Brownian motion. We also study the noise, in the sense of Tsirelson, generated by the flow. It is shown that this noise is not generated by any Brownian motion, even though it is predictable.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic flow, sticky Brownian motion, coalescence, stochastic differential equation, noise", } @Article{Atar:2002:SPC, author = "Rami Atar and Amarjit Budhiraja", title = "Stability Properties of Constrained Jump-Diffusion Processes", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "22:1--22:31", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-121", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/121", abstract = "We consider a class of jump-diffusion processes, constrained to a polyhedral cone $ G \subset \mathbb {R}^n $, where the constraint vector field is constant on each face of the boundary. The constraining mechanism corrects for ``attempts'' of the process to jump outside the domain. Under Lipschitz continuity of the Skorohod map $ \Gamma $, it is known that there is a cone $ {\cal C} $ such that the image $ \Gamma \phi $ of a deterministic linear trajectory $ \phi $ remains bounded if and only if $ \dot \phi \in {\cal C} $. Denoting the generator of a corresponding unconstrained jump-diffusion by $ \cal L $, we show that a key condition for the process to admit an invariant probability measure is that for $ x \in G $, $ {\cal L} \, {\rm id}(x) $ belongs to a compact subset of $ {\cal C}^o $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Jump diffusion processes. The Skorohod map. Stability cone. Harris recurrence", } @Article{Faure:2002:SNL, author = "Mathieu Faure", title = "Self-normalized Large Deviations for {Markov} Chains", journal = j-ELECTRON-J-PROBAB, volume = "7", pages = "23:1--23:31", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v7-122", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/122", abstract = "We prove a self-normalized large deviation principle for sums of Banach space valued functions of a Markov chain. Self-normalization applies to situations for which a full large deviation principle is not available. We follow the lead of Dembo and Shao [DemSha98b] who state partial large deviations principles for independent and identically distributed random sequences.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Large deviations, Markov chains, partial large deviation principles, self-normalization", } @Article{Dalang:2003:SNL, author = "Robert Dalang and Carl Mueller", title = "Some Non-Linear {S.P.D.E}'s That Are Second Order In Time", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "1:1--1:21", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-123", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/123", abstract = "We extend J. B. Walsh's theory of martingale measures in order to deal with stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert--Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic wave equation, stochastic beam equation, spatially homogeneous Gaussian noise, stochastic partial differential equations", } @Article{Hamadene:2003:RBS, author = "Said Hamad{\`e}ne and Youssef Ouknine", title = "Reflected Backward Stochastic Differential Equation with Jumps and Random Obstacle", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "2:1--2:20", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-124", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/124", abstract = "In this paper we give a solution for the one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process. We prove existence and uniqueness of the solution in using penalization and the Snell envelope theory. However both methods use a contraction in order to establish the result in the general case. Finally, we highlight the connection of such reflected BSDEs with integro-differential mixed stochastic optimal control.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Backward stochastic differential equation, penalization, Poisson point process, martingale representation theorem, integral-differential mixed control", } @Article{Cheridito:2003:FOU, author = "Patrick Cheridito and Hideyuki Kawaguchi and Makoto Maejima", title = "Fractional {Ornstein--Uhlenbeck} processes", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "3:1--3:14", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-125", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/125", abstract = "The classical stationary Ornstein--Uhlenbeck process can be obtained in two different ways. On the one hand, it is a stationary solution of the Langevin equation with Brownian motion noise. On the other hand, it can be obtained from Brownian motion by the so called Lamperti transformation. We show that the Langevin equation with fractional Brownian motion noise also has a stationary solution and that the decay of its auto-covariance function is like that of a power function. Contrary to that, the stationary process obtained from fractional Brownian motion by the Lamperti transformation has an auto-covariance function that decays exponentially.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Fractional Brownian motion, Langevin equation, Long-range dependence, Self-similar processes, Lamperti transformation", } @Article{Dawson:2003:SDM, author = "Donald Dawson and Andreas Greven", title = "State Dependent Multitype Spatial Branching Processes and their Longtime Behavior", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "4:1--4:93", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-126", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/126", abstract = "The paper focuses on spatial multitype branching systems with spatial components (colonies) indexed by a countable group, for example $ Z^d $ or the hierarchical group. As type space we allow continua and describe populations in one colony as measures on the type space. The spatial components of the system interact via migration. Instead of the classical independence assumption on the evolution of different families of the branching population, we introduce interaction between the families through a state dependent branching rate of individuals and in addition state dependent mean offspring of individuals. However for most results we consider the critical case in this work. The systems considered arise as diffusion limits of critical multiple type branching random walks on a countable group with interaction between individual families induced by a branching rate and offspring mean for a single particle, which depends on the total population at the site at which the particle in question is located.\par The main purpose of this paper is to construct the measure valued diffusions in question, characterize them via well-posed martingale problems and finally determine their longtime behavior, which includes some new features. Furthermore we determine the dynamics of two functionals of the system, namely the process of total masses at the sites and the relative weights of the different types in the colonies as system of interacting diffusions respectively time-inhomogeneous Fleming--Viot processes. This requires a detailed analysis of path properties of the total mass processes.\par In addition to the above mentioned systems of interacting measure valued processes we construct the corresponding historical processes via well-posed martingale problems. Historical processes include information on the family structure, that is, the varying degrees of relationship between individuals.\par Ergodic theorems are proved in the critical case for both the process and the historical process as well as the corresponding total mass and relative weights functionals. The longtime behavior differs qualitatively in the cases in which the symmetrized motion is recurrent respectively transient. We see local extinction in one case and honest equilibria in the other.\par This whole program requires the development of some new techniques, which should be of interest in a wider context. Such tools are dual processes in randomly fluctuating medium with singularities and coupling for systems with multi-dimensional components.\par The results above are the basis for the analysis of the large space-time scale behavior of such branching systems with interaction carried out in a forthcoming paper. In particular we study there the universality properties of the longtime behavior and of the family (or genealogical) structure, when viewed on large space and time scales.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Spatial branching processes with interaction, multitype branching processes with type-interaction, historical process, universality, coupling of multidimensional processes, coalescing random walks in singular random environment", } @Article{Kesten:2003:BRW, author = "Harry Kesten and Vladas Sidoravicius", title = "Branching Random Walk with Catalysts", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "5:1--5:51", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-127", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/127", abstract = "Shnerb et al. (2000), (2001) studied the following system of interacting particles on $ \mathbb {Z}^d $: There are two kinds of particles, called $A$-particles and $B$-particles. The $A$-particles perform continuous time simple random walks, independently of each other. The jump rate of each $A$-particle is $ D_A$. The $B$-particles perform continuous time simple random walks with jump rate $ D_B$, but in addition they die at rate $ \delta $ and a $B$-particle at $x$ at time $s$ splits into two particles at $x$ during the next $ d s$ time units with a probability $ \beta N_A(x, s)d s + o(d s)$, where $ N_A(x, s) \; (N_B(x, s))$ denotes the number of $A$-particles (respectively $B$-particles) at $x$ at time $s$. Conditionally on the $A$-system, the jumps, deaths and splittings of different $B$-particles are independent. Thus the $B$-particles perform a branching random walk, but with a birth rate of new particles which is proportional to the number of $A$-particles which coincide with the appropriate $B$-particles. One starts the process with all the $ N_A(x, 0), \, x \in \mathbb {Z}^d$, as independent Poisson variables with mean $ \mu_A$, and the $ N_B(x, 0), \, x \in \mathbb {Z}^d$, independent of the $A$-system, translation invariant and with mean $ \mu_B$.\par Shnerb et al. (2000) made the interesting discovery that in dimension 1 and 2 the expectation $ \mathbb {E} \{ N_B(x, t) \} $ tends to infinity, {\em no matter what the values of } $ \delta, \beta, D_A$, $ D_B, \mu_A, \mu_B \in (0, \infty)$ {\em are}. We shall show here that nevertheless {\em there is a phase transition in all dimensions}, that is, the system becomes (locally) extinct for large $ \delta $ but it survives for $ \beta $ large and $ \delta $ small.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Branching random walk, survival, extinction", } @Article{Sturm:2003:CPP, author = "Anja Sturm", title = "On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "6:1--6:39", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-129", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/129", abstract = "We consider the stochastic heat equation with a multiplicative colored noise term on the real space for dimensions greater or equal to 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Heat equation, colored noise, stochastic partial differential equation, superprocess, weak convergence, particle representation, random environment, existence theorem", } @Article{Bottcher:2003:NPL, author = "Albrecht B{\"o}ttcher and Sergei Grudsky", title = "The Norm of the Product of a Large Matrix and a Random Vector", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "7:1--7:29", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-132", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/132", abstract = "Given a real or complex $ n \times n $ matrix $ A_n $, we compute the expected value and the variance of the random variable $ \| A_n x \|^2 / \| A_n \|^2 $, where $x$ is uniformly distributed on the unit sphere of $ R^n$ or $ C^n$. The result is applied to several classes of structured matrices. It is in particular shown that if $ A_n$ is a Toeplitz matrix $ T_n(b)$, then for large $n$ the values of $ \| A_n x \| / \| A_n \| $ cluster fairly sharply around $ \| b \|_2 / \| b \|_\infty $ if $b$ is bounded and around zero in case $b$ is unbounded.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Condition number. Matrix norm. Random vector. Toeplitz matrix", } @Article{Fleischmann:2003:CSS, author = "Klaus Fleischmann and Leonid Mytnik", title = "Competing Species Superprocesses with Infinite Variance", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "8:1--8:59", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-136", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/136", abstract = "We study pairs of interacting measure-valued branching processes (superprocesses) with alpha-stable migration and $ (1 + \beta)$-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The competing species model describes the evolution of two populations where individuals of different types may kill each other if they collide. In the case of Brownian migration and finite variance branching, the model was introduced by Evans and Perkins (1994). The fact that now the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Superprocess with killing, competing superprocesses, interactive superprocesses, superprocess with immigration, measure-valued branching, interactive branching, state-dependent branching, collision measure, collision local time, martingale problem", } @Article{Bai:2003:BEB, author = "Zhi-Dong Bai and Hsien-Kuei Hwang and Tsung-Hsi Tsai", title = "{Berry--Ess{\'e}en} Bounds for the Number of Maxima in Planar Regions", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "9:1--9:26", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-137", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/137", abstract = "We derive the optimal convergence rate $ O(n^{-1 / 4}) $ in the central limit theorem for the number of maxima in random samples chosen uniformly at random from the right equilateral triangle with two sides parallel to the axes, the hypotenuse with the slope $ - 1 $ and constituting the top part of the boundary of the triangle. A local limit theorem with rate is also derived. The result is then applied to the number of maxima in general planar regions (upper-bounded by some smooth decreasing curves) for which a near-optimal convergence rate to the normal distribution is established.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Dominance, Maximal points, Central limit theorem, {Berry--Ess{\'e}en} bound, Local limit theorem, Method of moments", } @Article{Fitzsimmons:2003:HRM, author = "Patrick Fitzsimmons and Ronald Getoor", title = "Homogeneous Random Measures and Strongly Supermedian Kernels of a {Markov} Process", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "10:1--10:54", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-142", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/142", abstract = "The potential kernel of a positive {\em left} additive functional (of a strong Markov process $X$) maps positive functions to {\em strongly supermedian} functions and satisfies a variant of the classical {\em domination principle} of potential theory. Such a kernel $V$ is called a {\em regular strongly supermedian } kernel in recent work of L. Beznea and N. Boboc. We establish the converse: Every regular strongly supermedian kernel $V$ is the potential kernel of a random measure homogeneous on $ [0, \infty [$. Under additional finiteness conditions such random measures give rise to left additive functionals. We investigate such random measures, their potential kernels, and their associated characteristic measures. Given a left additive functional $A$ (not necessarily continuous), we give an explicit construction of a simple Markov process $Z$ whose resolvent has initial kernel equal to the potential kernel $ U_{\! A}$. The theory we develop is the probabilistic counterpart of the work of Beznea and Boboc. Our main tool is the Kuznetsov process associated with $X$ and a given excessive measure $m$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Homogeneous random measure, additive functional, Kuznetsov measure, potential kernel, characteristic measure, strongly supermedian, smooth measure", } @Article{Zhou:2003:CBC, author = "Xiaowen Zhou", title = "Clustering Behavior of a Continuous-Sites Stepping-Stone Model with {Brownian} Migration", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "11:1--11:15", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-141", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/141", abstract = "Clustering behavior is studied for a continuous-sites stepping-stone model with Brownian migration. It is shown that, if the model starts with the same mixture of different types of individuals over each site, then it will evolve in a way such that the site space is divided into disjoint intervals where only one type of individuals appear in each interval. Those intervals (clusters) are growing as time $t$ goes to infinity. The average size of the clusters at a fixed time $t$ is of the order of square root of $t$. Clusters at different times or sites are asymptotically independent as the difference of either the times or the sites goes to infinity.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "clustering; coalescing Brownian motion; stepping-stone model", } @Article{Marquez-Carreras:2003:LDP, author = "David Marquez-Carreras and Monica Sarra", title = "Large Deviation Principle for a Stochastic Heat Equation With Spatially Correlated Noise", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "12:1--12:39", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-146", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/146", abstract = "In this paper we prove a large deviation principle (LDP) for a perturbed stochastic heat equation defined on $ [0, T] \times [0, 1]^d $. This equation is driven by a Gaussian noise, white in time and correlated in space. Firstly, we show the Holder continuity for the solution of the stochastic heat equation. Secondly, we check that our Gaussian process satisfies an LDP and some requirements on the skeleton of the solution. Finally, we prove the called Freidlin--Wentzell inequality. In order to obtain all these results we need precise estimates of the fundamental solution of this equation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic partial differential equation, stochastic heat equation, Gaussian noise, large deviation principle", } @Article{Gao:2003:LTH, author = "Fuchang Gao and Jan Hannig and Tzong-Yow Lee and Fred Torcaso", title = "{Laplace} Transforms via {Hadamard} Factorization", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "13:1--13:20", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-151", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/151", abstract = "In this paper we consider the Laplace transforms of some random series, in particular, the random series derived as the squared $ L_2 $ norm of a Gaussian stochastic process. Except for some special cases, closed form expressions for Laplace transforms are, in general, rarely obtained. It is the purpose of this paper to show that for many Gaussian random processes the Laplace transform can be expressed in terms of well understood functions using complex-analytic theorems on infinite products, in particular, the Hadamard Factorization Theorem. Together with some tools from linear differential operators, we show that in many cases the Laplace transforms can be obtained with little work. We demonstrate this on several examples. Of course, once the Laplace transform is known explicitly one can easily calculate the corresponding exact $ L_2 $ small ball probabilities using Sytaja Tauberian theorem. Some generalizations are mentioned.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Small ball probability, Laplace Transforms, Hadamard's factorization theorem", } @Article{Tudor:2003:IFL, author = "Ciprian Tudor and Frederi Viens", title = "{It{\^o}} Formula and Local Time for the Fractional {Brownian} Sheet", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "14:1--14:31", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-155", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/155", abstract = "Using the techniques of the stochastic calculus of variations for Gaussian processes, we derive an It{\^o} formula for the fractional Brownian sheet with Hurst parameters bigger than $ 1 / 2 $. As an application, we give a stochastic integral representation for the local time of the fractional Brownian sheet.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "fractional Brownian sheet, It{\^o} formula, local time, Tanaka formula, Malliavin calculus", } @Article{Dembo:2003:BMC, author = "Amir Dembo and Yuval Peres and Jay Rosen", title = "{Brownian} Motion on Compact Manifolds: Cover Time and Late Points", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "15:1--15:14", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-139", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/139", abstract = "Let $M$ be a smooth, compact, connected Riemannian manifold of dimension $ d > 2$ and without boundary. Denote by $ T(x, r)$ the hitting time of the ball of radius $r$ centered at $x$ by Brownian motion on $M$. Then, $ C_r(M) = \sup_{x \in M} T(x, r)$ is the time it takes Brownian motion to come within $r$ of all points in $M$. We prove that $ C_r(M) / (r^{2 - d}| \log r|)$ tends to $ \gamma_d V(M)$ almost surely as $ r \to 0$, where $ V(M)$ is the Riemannian volume of $M$. We also obtain the ``multi-fractal spectrum'' $ f(\alpha)$ for ``late points'', i.e., the dimension of the set of $ \alpha $-late points $x$ in $M$ for which $ \limsup_{r \to 0} T(x, r) / (r^{2 - d}| \log r|) = \alpha > 0$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, manifold, cover time, Wiener sausage", } @Article{Budhiraja:2003:LDE, author = "Amarjit Budhiraja and Paul Dupuis", title = "Large Deviations for the Emprirical Measures of Reflecting {Brownian} Motion and Related Constrained Processes in {$ R_+ $}", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "16:1--16:46", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-154", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/154", abstract = "We consider the large deviations properties of the empirical measure for one dimensional constrained processes, such as reflecting Brownian motion, the M/M/1 queue, and discrete time analogues. Because these processes do not satisfy the strong stability assumptions that are usually assumed when studying the empirical measure, there is significant probability (from the perspective of large deviations) that the empirical measure charges the point at infinity. We prove the large deviation principle and identify the rate function for the empirical measure for these processes. No assumption of any kind is made with regard to the stability of the underlying process.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov process, constrained process, large deviations, empirical measure, stability, reflecting Brownian motion", } @Article{Delmas:2003:CML, author = "Jean-Fran{\c{c}}ois Delmas", title = "Computation of Moments for the Length of the One-Dimensional {ISE} Support", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "17:1--17:15", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-161", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/161", abstract = "We consider in this paper the support $ [L', R'] $ of the one dimensional Integrated Super Brownian Excursion. We determine the distribution of $ (R', L') $ through a modified Laplace transform. Then we give an explicit value for the first two moments of $ R' $ as well as the covariance of $ R' $ and $ {L'} $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian snake; ISE", } @Article{Gradinaru:2003:AFS, author = "Mihai Gradinaru and Ivan Nourdin", title = "Approximation at First and Second Order of $m$-order Integrals of the Fractional {Brownian} Motion and of Certain Semimartingales", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "18:1--18:26", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-166", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/166", abstract = "Let $X$ be the fractional Brownian motion of any Hurst index $ H \in (0, 1)$ (resp. a semimartingale) and set $ \alpha = H$ (resp. $ \alpha = \frac {1}{2}$). If $Y$ is a continuous process and if $m$ is a positive integer, we study the existence of the limit, as $ \varepsilon \rightarrow 0$, of the approximations\par $$ I_{\varepsilon }(Y, X) := \left \{ \int_0^t Y_s \left (\frac {X_{s + \varepsilon } - X_s}{\varepsilon^{\alpha }} \right)^m d s, \, t \geq 0 \right \} $$ of $m$-order integral of $Y$ with respect to $X$. For these two choices of $X$, we prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the $m$-th moment of the Gaussian standard random variable. In particular, if $m$ is an odd integer, the limit equals to zero. In this case, the convergence in distribution, as $ \varepsilon \rightarrow 0$, of $ \varepsilon^{- \frac {1}{2}} I_{\varepsilon }(1, X)$ is studied. We prove that the limit is a Brownian motion when $X$ is the fractional Brownian motion of index $ H \in (0, \frac {1}{2}]$, and it is in term of a two dimensional standard Brownian motion when $X$ is a semimartingale.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Maejima:2003:LMS, author = "Makoto Maejima and Kenji Yamamoto", title = "Long-Memory Stable {Ornstein--Uhlenbeck} Processes", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "19:1--19:18", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-168", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/168", abstract = "The solution of the Langevin equation driven by a L{\'e}vy process noise has been well studied, under the name of Ornstein--Uhlenbeck type process. It is a stationary Markov process. When the noise is fractional Brownian motion, the covariance of the stationary solution process has been studied by the first author with different coauthors. In the present paper, we consider the Langevin equation driven by a linear fractional stable motion noise, which is a selfsimilar process with long-range dependence but does not have finite variance, and we investigate the dependence structure of the solution process.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Lachal:2003:DST, author = "Aime Lachal", title = "Distributions of Sojourn Time, Maximum and Minimum for Pseudo-Processes Governed by Higher-Order Heat-Type Equations", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "20:1--20:53", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-178", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/178", abstract = "The higher-order heat-type equation $ \partial u / \partial t = \pm \partial^n u / \partial x^n $ has been investigated by many authors. With this equation is associated a pseudo-process $ (X_t)_{t \ge 0} $ which is governed by a signed measure. In the even-order case, Krylov (1960) proved that the classical arc-sine law of Paul Levy for standard Brownian motion holds for the pseudo-process $ (X_t)_{t \ge 0} $, that is, if $ T_t $ is the sojourn time of $ (X_t)_{t \ge 0} $ in the half line $ (0, + \infty) $ up to time $t$, then $ P(T_t \in d s) = \frac {ds}{\pi \sqrt {s(t - s)}}$, $ 0 < s < t$. Orsingher (1991) and next Hochberg and Orsingher (1994) obtained a counterpart to that law in the odd cases $ n = 3, 5, 7.$ Actually Hochberg and Orsingher (1994) proposed a more or less explicit expression for that new law in the odd-order general case and conjectured a quite simple formula for it. The distribution of $ T_t$ subject to some conditioning has also been studied by Nikitin \& Orsingher (2000) in the cases $ n = 3, 4.$ In this paper, we prove that the conjecture of Hochberg and Orsingher (1994) is true and we extend the results of Nikitin \& Orsingher for any integer $n$. We also investigate the distributions of maximal and minimal functionals of $ (X_t)_{t \ge 0}$, as well as the distribution of the last time before becoming definitively negative up to time $t$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Gao:2003:CTS, author = "Fuchang Gao and Jan Hannig and Fred Torcaso", title = "Comparison Theorems for Small Deviations of Random Series", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "21:1--21:17", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-147", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/147", abstract = "Let $ {\xi_n} $ be a sequence of i.i.d. positive random variables with common distribution function $ F(x) $. Let $ {a_n} $ and $ {b_n} $ be two positive non-increasing summable sequences such that $ {\prod_{n = 1}^{\infty }(a_n / b_n)} $ converges. Under some mild assumptions on $F$, we prove the following comparison\par $$ P \left (\sum_{n = 1}^{\infty }a_n \xi_n \leq \varepsilon \right) \sim \left (\prod_{n = 1}^{\infty } \frac {b_n}{a_n} \right)^{- \alpha } P \left (\sum_{n = 1}^{\infty }b_n \xi_n \leq \varepsilon \right), $$ where\par $$ { \alpha = \lim_{x \to \infty } \frac {\log F(1 / x)}{\log x}} < 0 $$ is the index of variation of $ F(1 / \cdot)$. When applied to the case $ \xi_n = |Z_n|^p$, where $ Z_n$ are independent standard Gaussian random variables, it affirms a conjecture of Li cite {Li1992a}.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "small deviation, random series, bounded variation", } @Article{Appleby:2003:EAS, author = "John Appleby and Alan Freeman", title = "Exponential Asymptotic Stability of Linear {It{\^o}--Volterra} Equation with Damped Stochastic Perturbations", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "22:1--22:22", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-179", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/179", abstract = "This paper studies the convergence rate of solutions of the linear It{\^o}-Volterra equation\par $$ d X(t) = \left (A X(t) + \int_0^t K(t - s)X(s), d s \right) \, d t + \Sigma (t) \, d W(t) \tag {1} $$ where $K$ and $ \Sigma $ are continuous matrix-valued functions defined on $ \mathbb {R}^+$, and $ (W(t))_{t \geq 0}$ is a finite-dimensional standard Brownian motion. It is shown that when the entries of $K$ are all of one sign on $ \mathbb {R}^+$, that (i) the almost sure exponential convergence of the solution to zero, (ii) the $p$-th mean exponential convergence of the solution to zero (for all $ p > 0$), and (iii) the exponential integrability of the entries of the kernel $K$, the exponential square integrability of the entries of noise term $ \Sigma $, and the uniform asymptotic stability of the solutions of the deterministic version of (1) are equivalent. The paper extends a result of Murakami which relates to the deterministic version of this problem.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Volkov:2003:ERW, author = "Stanislav Volkov", title = "Excited Random Walk on Trees", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "23:1--23:15", year = "2003", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-180", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/180", abstract = "We consider a nearest-neighbor stochastic process on a rooted tree $G$ which goes toward the root with probability $ 1 - \varepsilon $ when it visits a vertex for the first time. At all other times it behaves like a simple random walk on $G$. We show that for all $ \varepsilon \ge 0$ this process is transient. Also we consider a generalization of this process and establish its transience in {\em some} cases.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Ocone:2004:DVC, author = "Daniel Ocone and Ananda Weerasinghe", title = "Degenerate Variance Control in the One-dimensional Stationary Case", journal = j-ELECTRON-J-PROBAB, volume = "8", pages = "24:27", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v8-181", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/181", abstract = "We study the problem of stationary control by adaptive choice of the diffusion coefficient in the case that control degeneracy is allowed and the drift admits a unique, asymptotically stable equilibrium point. We characterize the optimal value and obtain it as an Abelian limit of optimal discounted values and as a limiting average of finite horizon optimal values, and we also characterize the optimal stationary strategy. In the case of linear drift, the optimal stationary value is expressed in terms of the solution of an optimal stopping problem. We generalize the above results to allow unbounded cost functions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stationary control, degenerate variance control; stochastic control", } @Article{Kozma:2004:AED, author = "Gady Kozma and Ehud Schreiber", title = "An asymptotic expansion for the discrete harmonic potential", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "1:1--1:17", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-170", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/170", abstract = "We give two algorithms that allow to get arbitrary precision asymptotics for the harmonic potential of a random walk.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Barbour:2004:NUB, author = "Andrew Barbour and Kwok Choi", title = "A non-uniform bound for translated {Poisson} approximation", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "2:18--2:36", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-182", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/182", abstract = "Let $ X_1, \ldots, X_n $ be independent, integer valued random variables, with $ p^{\text {th}} $ moments, $ p > 2 $, and let $W$ denote their sum. We prove bounds analogous to the classical non-uniform estimates of the error in the central limit theorem, but now, for approximation of $ {\cal L}(W)$ by a translated Poisson distribution. The advantage is that the error bounds, which are often of order no worse than in the classical case, measure the accuracy in terms of total variation distance. In order to have good approximation in this sense, it is necessary for $ {\cal L}(W)$ to be sufficiently smooth; this requirement is incorporated into the bounds by way of a parameter $ \alpha $, which measures the average overlap between $ {\cal L}(X_i)$ and $ {\cal L}(X_i + 1), 1 \leq i \leq n$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "non-uniform bounds; Stein's method; total variation; translated Poisson approximation", } @Article{Aldous:2004:BBA, author = "David Aldous and Gregory Miermont and Jim Pitman", title = "{Brownian} Bridge Asymptotics for Random $p$-Mappings", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "3:37--3:56", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-186", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/186", abstract = "The Joyal bijection between doubly-rooted trees and mappings can be lifted to a transformation on function space which takes tree-walks to mapping-walks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the Aldous--Pitman (1994) result on convergence of uniform random mapping walks to reflecting Brownian bridge, and extend this result to random $p$-mappings.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian bridge, Brownian excursion, Joyal map, random mapping, random tree, weak convergence", } @Article{Haas:2004:GSS, author = "B{\'e}n{\'e}dicte Haas and Gr{\'e}gory Miermont", title = "The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "4:57--4:97", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-187", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/187", abstract = "We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Continuum Random Tree is encoded in a normalized Brownian excursion. Under mild hypotheses, we then compute the Hausdorff dimensions of these trees, and the maximal H{\"o}lder exponents of the height functions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Mueller:2004:SPA, author = "Carl Mueller and Roger Tribe", title = "A Singular Parabolic {Anderson} Model", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "5:98--5:144", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-189", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/189", abstract = "We consider the heat equation with a singular random potential term. The potential is Gaussian with mean 0 and covariance given by a small constant times the inverse square of the distance. Solutions exist as singular measures, under suitable assumptions on the initial conditions and for sufficiently small noise. We investigate various properties of the solutions using such tools as scaling, self-duality and moment formulae. This model lies on the boundary between nonexistence and smooth solutions. It gives a new model, other than the superprocess, which has measure-valued solutions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic partial differential equations", } @Article{Fernandez:2004:CCC, author = "Roberto Fernandez and Gregory Maillard", title = "Chains with Complete Connections and One-Dimensional {Gibbs} Measures", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "6:145--6:176", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-149", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/149", abstract = "We discuss the relationship between one-dimensional Gibbs measures and discrete-time processes (chains). We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily Gibbsian).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Discrete-time processes, Chains with complete connections, Gibbs measures, Markov chains", } @Article{Ledoux:2004:DOS, author = "Michel Ledoux", title = "Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. {The} Continuous Case", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "7:177--7:208", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-191", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/191", abstract = "Following the investigation by U. Haagerup and S. Thorbjornsen, we present a simple differential approach to the limit theorems for empirical spectral distributions of complex random matrices from the Gaussian, Laguerre and Jacobi Unitary Ensembles. In the framework of abstract Markov diffusion operators, we derive by the integration by parts formula differential equations for Laplace transforms and recurrence equations for moments of eigenfunction measures. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. The moment recurrence relations are used to describe sharp, non asymptotic, small deviation inequalities on the largest eigenvalues at the rate given by the Tracy--Widom asymptotics.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Doney:2004:STB, author = "Ronald Doney", title = "Small-time Behaviour of {L{\'e}vy} Processes", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "8:209--8:229", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-193", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/193", abstract = "In this paper a neccessary and sufficient condition is established for the probability that a L{\'e}vy process is positive at time $t$ to tend to 1 as $t$ tends to 0. This condition is expressed in terms of the characteristics of the process, and is also shown to be equivalent to two probabilistic statements about the behaviour of the process for small time $t$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Alabert:2004:SDE, author = "Aureli Alabert and Miguel Angel Marmolejo", title = "Stochastic differential equations with boundary conditions driven by a {Poisson} noise", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "9:230--254", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-157", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/157", abstract = "We consider one-dimensional stochastic differential equations with a boundary condition, driven by a Poisson process. We study existence and uniqueness of solutions and the absolute continuity of the law of the solution. In the case when the coefficients are linear, we give an explicit form of the solution and study the reciprocal process property.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "boundary conditions; Poisson noise; reciprocal processes; stochastic differential equations", } @Article{Garet:2004:PTS, author = "Olivier Garet", title = "Percolation Transition for Some Excursion Sets", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "10:255--10:292", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-196", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/196", abstract = "We consider a random field $ (X_n)_{n \in \mathbb {Z}^d} $ and investigate when the set $ A_h = \{ k \in \mathbb {Z}^d; \vert X_k \vert \ge h \} $ has infinite clusters. The main problem is to decide whether the critical level\par $$ h_c = \sup \{ h \in R \colon P(A_h \text { has an infinite cluster }) > 0 \} $$ is neither $0$ nor $ + \infty $. Thus, we say that a percolation transition occurs. In a first time, we show that weakly dependent Gaussian fields satisfy to a well-known criterion implying the percolation transition. Then, we introduce a concept of percolation along reasonable paths and therefore prove a phenomenon of percolation transition for reasonable paths even for strongly dependent Gaussian fields. This allows to obtain some results of percolation transition for oriented percolation. Finally, we study some Gibbs states associated to a perturbation of a ferromagnetic quadratic interaction. At first, we show that a transition percolation occurs for superstable potentials. Next, we go to the critical case and show that a transition percolation occurs for directed percolation when $ d \ge 4$. We also note that the assumption of ferromagnetism can be relaxed when we deal with Gaussian Gibbs measures, i.e., when there is no perturbation of the quadratic interaction.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Kurkova:2004:ISC, author = "Irina Kurkova and Serguei Popov and M. Vachkovskaia", title = "On Infection Spreading and Competition between Independent Random Walks", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "11:293--11:315", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-197", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/197", abstract = "We study the models of competition and spreading of infection for infinite systems of independent random walks. For the competition model, we investigate the question whether one of the spins prevails with probability one. For the infection spreading, we give sufficient conditions for recurrence and transience (i.e., whether the origin will be visited by infected particles infinitely often a.s.).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Dawson:2004:HEB, author = "Donald Dawson and Luis Gorostiza and Anton Wakolbinger", title = "Hierarchical Equilibria of Branching Populations", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "12:316--12:381", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-200", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/200", abstract = "The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group $ \Omega_N $ consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit $ N \to \infty $ (called the {\em hierarchical mean field limit}), the equilibrium aggregated populations in a nested sequence of balls $ B^{(N)}_\ell $ of hierarchical radius $ \ell $ converge to a backward Markov chain on $ \mathbb {R_+} $. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Multilevel branching, hierarchical mean-field limit, strong transience, genealogy", } @Article{Kendall:2004:CIK, author = "Wilfrid Kendall and Catherine Price", title = "Coupling Iterated {Kolmogorov} Diffusions", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "13:382--13:410", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-201", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/201", abstract = "The {\em Kolmogorov-1934 diffusion} is the two-dimensional diffusion generated by real Brownian motion and its time integral. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals as further components to the original Kolmogorov diffusion. A Laplace-transform argument shows it is not possible successfully to couple all iterated time integrals at once; however we give an explicit construction of a successful co-adapted coupling method for Brownian motion, its time integral, and its twice-iterated time integral; and a more implicit construction of a successful co-adapted coupling method which works for finite sets of iterated time integrals.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{vonRenesse:2004:ICR, author = "Max-K. von Renesse", title = "Intrinsic Coupling on {Riemannian} Manifolds and Polyhedra", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "14:411--14:435", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-205", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/205", abstract = "Starting from a central limit theorem for geometric random walks we give an elementary construction of couplings between Brownian motions on Riemannian manifolds. This approach shows that cut locus phenomena are indeed inessential for Kendall's and Cranston's stochastic proof of gradient estimates for harmonic functions on Riemannian manifolds with lower curvature bounds. Moreover, since the method is based on an asymptotic quadruple inequality and a central limit theorem only it may be extended to certain non smooth spaces which we illustrate by the example of Riemannian polyhedra. Here we also recover the classical heat kernel gradient estimate which is well known from the smooth setting.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Central Limit Theorem; Coupling; Gradient Estimates", } @Article{Loewe:2004:RMR, author = "Matthias Loewe and Heinrich Matzinger and Franz Merkl", title = "Reconstructing a Multicolor Random Scenery seen along a Random Walk Path with Bounded Jumps", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "15:436--15:507", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-206", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/206", abstract = "Kesten noticed that the scenery reconstruction method proposed by Matzinger in his PhD thesis relies heavily on the skip-free property of the random walk. He asked if one can still reconstruct an i.i.d. scenery seen along the path of a non-skip-free random walk. In this article, we positively answer this question. We prove that if there are enough colors and if the random walk is recurrent with at most bounded jumps, and if it can reach every integer, then one can almost surely reconstruct almost every scenery up to translations and reflections. Our reconstruction method works if there are more colors in the scenery than possible single steps for the random walk.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "ergodic theory; jumps; random walk; Scenery reconstruction; stationary processes", } @Article{Barral:2004:MAC, author = "Julien Barral and Jacques V{\'e}hel", title = "Multifractal Analysis of a Class of Additive Processes with Correlated Non-Stationary Increments", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "16:508--16:543", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-208", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/208", abstract = "We consider a family of stochastic processes built from infinite sums of independent positive random functions on $ R_+ $. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on $ R_+ $. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic on the Internet. Such processes bear some analogy with L{\'e}vy processes, but they are more complex in the sense that their increments are neither stationary nor independent. Nevertheless, we show that their multifractal behavior is very much the same as that of certain L{\'e}vy processes. More precisely, we compute the Hausdorff multifractal spectrum of our processes, and find that it shares the shape of the spectrum of a typical L{\'e}vy process. This result yields a theoretical basis to the empirical discovery of the multifractal nature of TCP traffic.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Shao:2004:ADB, author = "Qi-Man Shao and Chun Su and Gang Wei", title = "Asymptotic Distributions and {Berry--Ess{\'e}en} Bounds for Sums of Record Values", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "17:544--17:559", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-210", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/210", abstract = "Let $ \{ U_n, n \geq 1 \} $ be independent uniformly distributed random variables, and $ \{ Y_n, n \geq 1 \} $ be independent and identically distributed non-negative random variables with finite third moments. Denote $ S_n = \sum_{i = 1}^n Y_i $ and assume that $ (U_1, \cdots, U_n) $ and $ S_{n + 1} $ are independent for every fixed $n$. In this paper we obtain {Berry--Ess{\'e}en} bounds for $ \sum_{i = 1}^n \psi (U_i S_{n + 1})$, where $ \psi $ is a non-negative function. As an application, we give {Berry--Ess{\'e}en} bounds and asymptotic distributions for sums of record values.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Kouritzin:2004:NFR, author = "Michael Kouritzin and Wei Sun and Jie Xiong", title = "Nonliner Filtering for Reflecting Diffusions in Random Environments via Nonparametric Estimation", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "18:560--18:574", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-214", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See erratum \cite{Kouritzin:2017:ENF}.", URL = "http://ejp.ejpecp.org/article/view/214", abstract = "We study a nonlinear filtering problem in which the signal to be estimated is a reflecting diffusion in a random environment. Under the assumption that the observation noise is independent of the signal, we develop a nonparametric functional estimation method for finding workable approximate solutions to the conditional distributions of the signal state. Furthermore, we show that the pathwise average distance, per unit time, of the approximate filter from the optimal filter is asymptotically small in time. Also, we use simulations based upon a particle filter algorithm to show the efficiency of the method.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Bertoin:2004:ALN, author = "Jean Bertoin and Alexander Gnedin", title = "Asymptotic Laws for Nonconservative Self-similar Fragmentations", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "19:575--19:593", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-215", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/215", abstract = "We consider a self-similar fragmentation process in which the generic particle of mass $x$ is replaced by the offspring particles at probability rate $ x^\alpha $, with positive parameter $ \alpha $. The total of offspring masses may be both larger or smaller than $x$ with positive probability. We show that under certain conditions the typical mass in the ensemble is of the order $ t^{-1 / \alpha }$ and that the empirical distribution of masses converges to a random limit which we characterise in terms of the reproduction law.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Nualart:2004:LSM, author = "Eulalia Nualart and Thomas Mountford", title = "Level Sets of Multiparameter {Brownian} Motions", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "20:594--20:614", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-169", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/169", abstract = "We use Girsanov's theorem to establish a conjecture of Khoshnevisan, Xiao and Zhong that $ \phi (r) = r^{N - d / 2} (\log \log (\frac {1}{r}))^{d / 2} $ is the exact Hausdorff measure function for the zero level set of an $N$-parameter $d$-dimensional additive Brownian motion. We extend this result to a natural multiparameter version of Taylor and Wendel's theorem on the relationship between Brownian local time and the Hausdorff $ \phi $-measure of the zero set.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "additive Brownian motion; Hausdorff measure; level sets; Local times", } @Article{Krylov:2004:QIS, author = "N. V. Krylov", title = "Quasiderivatives and Interior Smoothness of Harmonic Functions Associated with Degenerate Diffusion Processes", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "21:615--21:633", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-219", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/219", abstract = "Proofs and two applications of two general results are given concerning the problem of establishing interior smoothness of probabilistic solutions of elliptic degenerate equations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Bass:2004:CSD, author = "Richard Bass and Edwin Perkins", title = "Countable Systems of Degenerate Stochastic Differential Equations with Applications to Super-{Markov} Chains", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "22:634--22:673", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-222", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/222", abstract = "We prove well-posedness of the martingale problem for an infinite-dimensional degenerate elliptic operator under appropriate H{\"o}lder continuity conditions on the coefficients. These martingale problems include large population limits of branching particle systems on a countable state space in which the particle dynamics and branching rates may depend on the entire population in a H{\"o}lder fashion. This extends an approach originally used by the authors in finite dimensions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Denis:2004:GAR, author = "Laurent Denis and L. Stoica", title = "A General Analytical Result for Non-linear {SPDE}'s and Applications", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "23:674--23:709", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-223", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/223", abstract = "Using analytical methods, we prove existence uniqueness and estimates for s.p.d.e. of the type\par $$ d u_t + A u_t d t + f (t, u_t) d t + R g(t, u_t) d t = h(t, x, u_t) d B_t, $$ where $A$ is a linear non-negative self-adjoint (unbounded) operator, $f$ is a nonlinear function which depends on $u$ and its derivatives controlled by $ \sqrt {A} u$, $ R g$ corresponds to a nonlinearity involving $u$ and its derivatives of the same order as $ A u$ but of smaller magnitude, and the right term contains a noise involving a $d$-dimensional Brownian motion multiplied by a non-linear function. We give a neat condition concerning the magnitude of these nonlinear perturbations. We also mention a few examples and, in the case of a diffusion generator, we give a double stochastic interpretation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{vanderHofstad:2004:GSC, author = "Remco van der Hofstad and Akira Sakai", title = "{Gaussian} Scaling for the Critical Spread-out Contact Process above the Upper Critical Dimension", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "24:710--24:769", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-224", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/224", abstract = "We consider the critical spread-out contact process in $ Z^d $ with $ d \geq 1 $, whose infection range is denoted by $ L \geq 1 $. The two-point function $ \tau_t(x) $ is the probability that $ x \in Z^d $ is infected at time $t$ by the infected individual located at the origin $ o \in Z^d$ at time 0. We prove Gaussian behaviour for the two-point function with $ L \geq L_0$ for some finite $ L_0 = L_0 (d)$ for $ d > 4$. When $ d \leq 4$, we also perform a local mean-field limit to obtain Gaussian behaviour for $ \tau_{ tT}(x)$ with $ t > 0$ fixed and $ T \to \infty $ when the infection range depends on $T$ in such a way that $ L_T = L T^b$ for any $ b > (4 - d) / 2 d$.\par The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process. We prove the existence of several critical exponents and show that they take on their respective mean-field values. The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards those for the canonical measure of super-Brownian motion, which we defer to a sequel of this paper.\par The results in this paper also apply to oriented percolation, for which we reprove some of the results in \cite{hs01} and extend the results to the local mean-field setting described above when $ d \leq 4$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Berestycki:2004:EFC, author = "Julien Berestycki", title = "Exchangeable Fragmentation--Coalescence Processes and their Equilibrium Measures", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "25:770--25:824", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-227", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/227", abstract = "We define and study a family of Markov processes with state space the compact set of all partitions of $N$ that we call exchangeable fragmentation-coalescence processes. They can be viewed as a combination of homogeneous fragmentation as defined by Bertoin and of homogeneous coalescence as defined by Pitman and Schweinsberg or M{\"o}hle and Sagitov. We show that they admit a unique invariant probability measure and we study some properties of their paths and of their equilibrium measure.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Peres:2004:MTR, author = "Yuval Peres and David Revelle", title = "Mixing Times for Random Walks on Finite Lamplighter Groups", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "26:825--26:845", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-198", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/198", abstract = "Given a finite graph $G$, a vertex of the lamplighter graph $ G^\diamondsuit = \mathbb {Z}_2 \wr G$ consists of a zero-one labeling of the vertices of $G$, and a marked vertex of $G$. For transitive $G$ we show that, up to constants, the relaxation time for simple random walk in $ G^\diamondsuit $ is the maximal hitting time for simple random walk in $G$, while the mixing time in total variation on $ G^\diamondsuit $ is the expected cover time on $G$. The mixing time in the uniform metric on $ G^\diamondsuit $ admits a sharp threshold, and equals $ |G|$ multiplied by the relaxation time on $G$, up to a factor of $ \log |G|$. For $ \mathbb {Z}_2 \wr \mathbb {Z}_n^2$, the lamplighter group over the discrete two dimensional torus, the relaxation time is of order $ n^2 \log n$, the total variation mixing time is of order $ n^2 \log^2 n$, and the uniform mixing time is of order $ n^4$. For $ \mathbb {Z}_2 \wr \mathbb {Z}_n^d$ when $ d \geq 3$, the relaxation time is of order $ n^d$, the total variation mixing time is of order $ n^d \log n$, and the uniform mixing time is of order $ n^{d + 2}$. In particular, these three quantities are of different orders of magnitude.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "cover time; lamplighter group; mixing time; random walks", } @Article{Lawler:2004:BEC, author = "Gregory Lawler and Vlada Limic", title = "The {Beurling} Estimate for a Class of Random Walks", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "27:846--27:861", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-228", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/228", abstract = "An estimate of Beurling states that if $K$ is a curve from $0$ to the unit circle in the complex plane, then the probability that a Brownian motion starting at $ - \varepsilon $ reaches the unit circle without hitting the curve is bounded above by $ c \varepsilon^{1 / 2}$. This estimate is very useful in analysis of boundary behavior of conformal maps, especially for connected but rough boundaries. The corresponding estimate for simple random walk was first proved by Kesten. In this note we extend this estimate to random walks with zero mean, finite $ (3 + \delta)$-moment.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Beurling projection; escape probabilities; Green's function; random walk", } @Article{Puhalskii:2004:SDL, author = "Anatolii Puhalskii", title = "On Some Degenerate Large Deviation Problems", journal = j-ELECTRON-J-PROBAB, volume = "9", pages = "28:862--28:886", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v9-232", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/232", abstract = "This paper concerns the issue of obtaining the large deviation principle for solutions of stochastic equations with possibly degenerate coefficients. Specifically, we explore the potential of the methodology that consists in establishing exponential tightness and identifying the action functional via a maxingale problem. In the author's earlier work it has been demonstrated that certain convergence properties of the predictable characteristics of semimartingales ensure both that exponential tightness holds and that every large deviation accumulation point is a solution to a maxingale problem. The focus here is on the uniqueness for the maxingale problem. It is first shown that under certain continuity hypotheses existence and uniqueness of a solution to a maxingale problem of diffusion type are equivalent to Luzin weak existence and uniqueness, respectively, for the associated idempotent It{\^o} equation. Consequently, if the idempotent equation has a unique Luzin weak solution, then the action functional is specified uniquely, so the large deviation principle follows. Two kinds of application are considered. Firstly, we obtain results on the logarithmic asymptotics of moderate deviations for stochastic equations with possibly degenerate diffusion coefficients which, as compared with earlier results, relax the growth conditions on the coefficients, permit certain non-Lipshitz-continuous coefficients, and allow the coefficients to depend on the entire past of the process and to be discontinuous functions of time. The other application concerns multiple-server queues with impatient customers.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Kim:2005:ESD, author = "Kyeong-Hun Kim", title = "{$ L_p $}-Estimates for {SPDE} with Discontinuous Coefficients in Domains", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "1:1--1:20", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-234", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/234", abstract = "Stochastic partial differential equations of divergence form with discontinuous and unbounded coefficients are considered in $ C^1 $ domains. Existence and uniqueness results are given in weighted $ L_p $ spaces, and Holder type estimates are presented.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic partial differential equations, discontinuous coefficients", } @Article{Newman:2005:CCN, author = "Charles Newman and Krishnamurthi Ravishankar and Rongfeng Sun", title = "Convergence of Coalescing Nonsimple Random Walks to the {Brownian Web}", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "2:21--2:60", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-235", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/235", abstract = "The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $ R \times R $. It was first introduced by Arratia, and later analyzed in detail by Toth and Werner. More recently, Fontes, Isopi, Newman and Ravishankar (FINR) gave a characterization of the BW, and general convergence criteria allowing in principle either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. To date, the general convergence criteria of FINR have not been verified for any case with crossing paths, which appears to be significantly more difficult than the noncrossing paths case. Accordingly, in this paper, we formulate new convergence criteria for the crossing paths case, and verify them for non-simple coalescing random walks satisfying a finite fifth moment condition. This is the first time that convergence to the BW has been proved for models with crossing paths. Several corollaries are presented, including an analysis of the scaling limit of voter model interfaces that extends a result of Cox and Durrett.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian Web, Invariance Principle, Coalescing Random Walks, Brownian Networks, Continuum Limit", } @Article{Kontoyiannis:2005:LDA, author = "Ioannis Kontoyiannis and Sean Meyn", title = "Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular {Markov} Processes", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "3:61--3:123", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-231", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/231", abstract = "In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker--Varadhan. For any such process $ \{ \Phi (t) \} $ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $ \hat P(x, d y) = e^{F(x)} P(x, d y)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a ``maximal, '' well-behaved solution to the ``multiplicative Poisson equation, '' defined as an eigenvalue problem for $ \hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker--Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov process, large deviations, entropy, Lyapunov function, empirical measures, nonlinear generator, large deviations principle", } @Article{Bass:2005:ASI, author = "Richard Bass and Jay Rosen", title = "An Almost Sure Invariance Principle for Renormalized Intersection Local Times", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "4:124--4:164", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-236", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/236", abstract = "Let $ \beta_k(n) $ be the number of self-intersections of order $k$, appropriately renormalized, for a mean zero planar random walk with $ 2 + \delta $ moments. On a suitable probability space we can construct the random walk and a planar Brownian motion $ W_t$ such that for each $ k \geq 2$, $ | \beta_k(n) - \gamma_k(n)| = o(1)$, a.s., where $ \gamma_k(n)$ is the renormalized self-intersection local time of order $k$ at time 1 for the Brownian motion $ W_{nt} / \sqrt n$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Schuhmacher:2005:DEP, author = "Dominic Schuhmacher", title = "Distance Estimates for {Poisson} Process Approximations of Dependent Thinnings", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "5:165--5:201", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-237", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/237", abstract = "It is well known, that under certain conditions, gradual thinning of a point process on $ R^d_+ $, accompanied by a contraction of space to compensate for the thinning, leads in the weak limit to a Cox process. In this article, we apply discretization and a result based on Stein's method to give estimates of the Barbour--Brown distance $ d_2 $ between the distribution of a thinned point process and an approximating Poisson process, and evaluate the estimates in concrete examples. We work in terms of two, somewhat different, thinning models. The main model is based on the usual thinning notion of deleting points independently according to probabilities supplied by a random field. In Section 4, however, we use an alternative thinning model, which can be more straightforward to apply if the thinning is determined by point interactions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Eisenbaum:2005:CBG, author = "Nathalie Eisenbaum", title = "A Connection between {Gaussian} Processes and {Markov} Processes", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "6:202--6:215", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-238", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/238", abstract = "The Green function of a transient symmetric Markov process can be interpreted as the covariance of a centered Gaussian process. This relation leads to several fruitful identities in law. Symmetric Markov processes and their associated Gaussian process both benefit from these connections. Therefore it is of interest to characterize the associated Gaussian processes. We present here an answer to that question.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Cancrini:2005:DLT, author = "Nicoletta Cancrini and Filippo Cesi and Cyril Roberto", title = "Diffusive Long-time Behavior of {Kawasaki} Dynamics", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "7:216--7:249", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-239", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/239", abstract = "If $ P_t $ is the semigroup associated with the Kawasaki dynamics on $ Z^d $ and $f$ is a local function on the configuration space, then the variance with respect to the invariant measure $ \mu $ of $ P_t f$ goes to zero as $ t \to \infty $ faster than $ t^{-d / 2 + \varepsilon }$, with $ \varepsilon $ arbitrarily small. The fundamental assumption is a mixing condition on the interaction of Dobrushin and Schlosman type.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Heicklen:2005:RPS, author = "Deborah Heicklen and Christopher Hoffman", title = "Return Probabilities of a Simple Random Walk on Percolation Clusters", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "8:250--8:302", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-240", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/240", abstract = "We bound the probability that a continuous time simple random walk on the infinite percolation cluster on $ Z^d $ returns to the origin at time $t$. We use this result to show that in dimensions 5 and higher the uniform spanning forest on infinite percolation clusters supported on graphs with infinitely many connected components a.s.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Birkner:2005:ASB, author = "Matthias Birkner and Jochen Blath and Marcella Capaldo and Alison Etheridge and Martin M{\"o}hle and Jason Schweinsberg and Anton Wakolbinger", title = "Alpha-Stable Branching and Beta-Coalescents", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "9:303--9:325", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-241", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/241", abstract = "We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $ \alpha $-stable branching mechanisms. The random ancestral partition is then a time-changed $ \Lambda $-coalescent, where $ \Lambda $ is the Beta-distribution with parameters $ 2 - \alpha $ and $ \alpha $, and the time change is given by $ Z^{1 - \alpha }$, where $Z$ is the total population size. For $ \alpha = 2$ (Feller's branching diffusion) and $ \Lambda = \delta_0$ (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem. For $ \alpha = 1$ and $ \Lambda $ the uniform distribution on $ [0, 1]$, this is the duality discovered by Bertoin \& Le Gall (2000) between the norming of Neveu's continuous state branching process and the Bolthausen--Sznitman coalescent.\par We present two approaches: one, exploiting the `modified lookdown construction', draws heavily on Donnelly \& Kurtz (1999); the other is based on direct calculations with generators.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Berzin:2005:CFM, author = "Corinne Berzin and Jos{\'e} Le{\'o}n", title = "Convergence in Fractional Models and Applications", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "10:326--10:370", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-172", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/172", abstract = "We consider a fractional Brownian motion with Hurst parameter strictly between 0 and 1. We are interested in the asymptotic behaviour of functionals of the increments of this and related processes and we propose several probabilistic and statistical applications.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "fractional Brownian motion; Level crossings; limit theorem; local time; rate of convergence", } @Article{Salminen:2005:PIF, author = "Paavo Salminen and Marc Yor", title = "Perpetual Integral Functionals as Hitting and Occupation Times", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "11:371--11:419", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-256", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/256", abstract = "Let $X$ be a linear diffusion and $f$ a non-negative, Borel measurable function. We are interested in finding conditions on $X$ and $f$ which imply that the perpetual integral functional\par $$ I^X_\infty (f) := \int_0^\infty f(X_t) d t $$ is identical in law with the first hitting time of a point for some other diffusion. This phenomenon may often be explained using random time change. Because of some potential applications in mathematical finance, we are considering mainly the case when $X$ is a Brownian motion with drift $ \mu > 0, $ denoted $ {B^{(\mu)}_t \colon t \geq 0}, $ but it is obvious that the method presented is more general. We also review the known examples and give new ones. In particular, results concerning one-sided functionals\par $$ \int_0^\infty f(B^{(\mu)}_t){\bf 1}_{{B^{(\mu)}_t < 0}} d t \quad {\rm and} \quad \int_0^\infty f(B^{(\mu)}_t){\bf 1}_{{B^{(\mu)}_t > 0}} d t $$ are presented. This approach generalizes the proof, based on the random time change techniques, of the fact that the Dufresne functional (this corresponds to $ f(x) = \exp ( - 2 x)), $ playing quite an important role in the study of geometric Brownian motion, is identical in law with the first hitting time for a Bessel process. Another functional arising naturally in this context is\par $$ \int_0^\infty \big (a + \exp (B^{(\mu)}_t) \big)^{-2} d t, $$ which is seen, in the case $ \mu = 1 / 2, $ to be identical in law with the first hitting time for a Brownian motion with drift $ \mu = a / 2.$ The paper is concluded by discussing how the Feynman--Kac formula can be used to find the distribution of a perpetual integral functional.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Chauvin:2005:MPB, author = "B. Chauvin and T. Klein and J.-F. Marckert and A. Rouault", title = "Martingales and Profile of Binary Search Trees", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "12:420--12:435", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-257", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/257", abstract = "We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Mountford:2005:TCN, author = "Thomas Mountford and Li-Chau Wu", title = "The Time for a Critical Nearest Particle System to reach Equilibrium starting with a large Gap", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "13:436--13:498", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-242", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/242", abstract = "We consider the time for a critical nearest particle system, starting in equilibrium subject to possessing a large gap, to achieve equilibrium.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Interacting Particle Systems, Reversibility, Convergence to equilibrium", } @Article{Panchenko:2005:CLT, author = "Dmitry Panchenko", title = "A {Central Limit Theorem} for Weighted Averages of Spins in the High Temperature Region of the {Sherrington--Kirkpatrick} Model", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "14:499--14:524", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-258", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/258", abstract = "In this paper we prove that in the high temperature region of the Sherrington--Kirkpatrick model for a typical realization of the disorder the weighted average of spins $ \sum_{i \leq N} t_i \sigma_i $ will be approximately Gaussian provided that $ \max_{i \leq N}|t_i| / \sum_{i \leq N} t_i^2 $ is small.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{DaiPra:2005:LSI, author = "Paolo {Dai Pra} and Gustavo Posta", title = "Logarithmic {Sobolev} Inequality for Zero--Range Dynamics: Independence of the Number of Particles", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "15:525--15:576", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-259", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/259", abstract = "We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a box of diameter $L$ may depend on $L$ but not on the number of particles. This is a first, but relevant and quite technical step, in the proof that this logarithmic-Sobolev constant grows as the square of $L$, that is presented in a forthcoming paper.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Chen:2005:LDL, author = "Xia Chen and Wenbo Li and Jay Rosen", title = "Large Deviations for Local Times of Stable Processes and Stable Random Walks in 1 Dimension", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "16:577--16:608", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-260", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/260", abstract = "In Chen and Li (2004), large deviations were obtained for the spatial $ L^p $ norms of products of independent Brownian local times and local times of random walks with finite second moment. The methods of that paper depended heavily on the continuity of the Brownian path and the fact that the generator of Brownian motion, the Laplacian, is a local operator. In this paper we generalize these results to local times of symmetric stable processes and stable random walks.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Biggins:2005:FPS, author = "John Biggins and Andreas Kyprianou", title = "Fixed Points of the Smoothing Transform: the Boundary Case", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "17:609--17:631", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-255", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/255", abstract = "Let $ A = (A_1, A_2, A_3, \ldots) $ be a random sequence of non-negative numbers that are ultimately zero with $ E[\sum A_i] = 1 $ and $ E \left [\sum A_i \log A_i \right] \leq 0 $. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation $ \Phi (\psi) = E \left [\prod_i \Phi (\psi A_i) \right], $ where $ \Phi $ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when $ E \left [\sum A_i \log A_i \right] < 0 $. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where $ E \left [\sum A_i \log A_i \right] = 0 $, are obtained.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "branching random walk; functional equation; Smoothing transform", } @Article{Cabanal-Duvillard:2005:MRB, author = "Thierry Cabanal-Duvillard", title = "A Matrix Representation of the {Bercovici--Pata} Bijection", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "18:632--18:661", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-246", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/246", abstract = "Let $ \mu $ be an infinitely divisible law on the real line, $ \Lambda (\mu) $ its freely infinitely divisible image by the Bercovici--Pata bijection. The purpose of this article is to produce a new kind of random matrices with distribution $ \mu $ at dimension 1, and with its empirical spectral law converging to $ \Lambda (\mu) $ as the dimension tends to infinity. This constitutes a generalisation of Wigner's result for the Gaussian Unitary Ensemble.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random matrices, free probability, infinitely divisible laws", } @Article{Lozada-Chang:2005:LDM, author = "Li-Vang Lozada-Chang", title = "Large Deviations on Moment Spaces", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "19:662--19:690", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-202", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/202", abstract = "In this paper we study asymptotic behavior of some moment spaces. We consider two different settings. In the first one, we work with ordinary multi-dimensional moments on the standard $m$-simplex. In the second one, we deal with the trigonometric moments on the unit circle of the complex plane. We state large and moderate deviation principles for uniformly distributed moments. In both cases the rate function of the large deviation principle is related to the reversed Kullback information with respect to the uniform measure on the integration space.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "large deviations; multidimensional moment; random moment problem", } @Article{Begyn:2005:QVA, author = "Arnaud Begyn", title = "Quadratic Variations along Irregular Subdivisions for {Gaussian} Processes", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "20:691--20:717", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-245", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/245", abstract = "In this paper we deal with second order quadratic variations along general subdivisions for processes with Gaussian increments. These have almost surely a deterministic limit under conditions on the mesh of the subdivisions. This limit depends on the singularity function of the process and on the structure of the subdivisions too. Then we illustrate the results with the example of the time-space deformed fractional Brownian motion and we present some simulations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "estimation, fractional processes, Gaussian processes, generalized quadratic variations, irregular subdivisions, singularity function", } @Article{Goldschmidt:2005:RRT, author = "Christina Goldschmidt and James Martin", title = "Random Recursive Trees and the {Bolthausen--Sznitman} Coalesent", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "21:718--21:745", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-265", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/265", abstract = "We describe a representation of the Bolthausen--Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to $ [n] $: we show that the distribution of the number of blocks involved in the final collision converges as $ n \to \infty $, and obtain a scaling law for the sizes of these blocks. We also consider the discrete-time Markov chain giving the number of blocks after each collision of the coalescent restricted to $ [n] $; we show that the transition probabilities of the time-reversal of this Markov chain have limits as $ n \to \infty $. These results can be interpreted as describing a ``post-gelation'' phase of the Bolthausen--Sznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Bouchard:2005:HAO, author = "Bruno Bouchard and Emmanuel Teman", title = "On the Hedging of {American} Options in Discrete Time with Proportional Transaction Costs", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "22:746--22:760", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-266", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/266", abstract = "In this note, we consider a general discrete time financial market with proportional transaction costs as in Kabanov and Stricker (2001), Kabanov et al. (2002), Kabanov et al. (2003) and Schachermayer (2004). We provide a dual formulation for the set of initial endowments which allow to super-hedge some American claim. We show that this extends the result of Chalasani and Jha (2001) which was obtained in a model with constant transaction costs and risky assets which evolve on a finite dimensional tree. We also provide fairly general conditions under which the expected formulation in terms of stopping times does not work.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Coutin:2005:SMR, author = "Laure Coutin and Antoine Lejay", title = "Semi-martingales and rough paths theory", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "23:761--23:785", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-162", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/162", abstract = "We prove that the theory of rough paths, which is used to define path-wise integrals and path-wise differential equations, can be used with continuous semi-martingales. We provide then an almost sure theorem of type Wong--Zakai. Moreover, we show that the conditions UT and UCV, used to prove that one can interchange limits and It{\^o} or Stratonovich integrals, provide the same result when one uses the rough paths theory.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "$p$-variation; conditions UT and UCV; iterated integrals; rough paths; Semi-martingales; Wong--Zakai theorem", } @Article{Cassandro:2005:ODR, author = "Marzio Cassandro and Enza Orlandi and Pierre Picco and Maria Eulalia Vares", title = "One-dimensional Random Field {Kac}'s Model: Localization of the Phases", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "24:786--24:864", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-263", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/263", abstract = "We study the typical profiles of a one dimensional random field Kac model, for values of the temperature and magnitude of the field in the region of two absolute minima for the free energy of the corresponding random field Curie Weiss model. We show that, for a set of realizations of the random field of overwhelming probability, the localization of the two phases corresponding to the previous minima is completely determined. Namely, we are able to construct random intervals tagged with a sign, where typically, with respect to the infinite volume Gibbs measure, the profile is rigid and takes, according to the sign, one of the two values corresponding to the previous minima. Moreover, we characterize the transition from one phase to the other. The analysis extends the one done by Cassandro, Orlandi and Picco in [13].", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Phase transition, random walk, random environment, Kac potential", } @Article{Flandoli:2005:SVF, author = "Franco Flandoli and Massimiliano Gubinelli", title = "Statistics of a Vortex Filament Model", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "25:865--25:900", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-267", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/267", abstract = "A random incompressible velocity field in three dimensions composed by Poisson distributed Brownian vortex filaments is constructed. The filaments have a random thickness, length and intensity, governed by a measure $ \gamma $. Under appropriate assumptions on $ \gamma $ we compute the scaling law of the structure function of the field and show that, in particular, it allows for either K41-like scaling or multifractal scaling.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Fulman:2005:SMD, author = "Jason Fulman", title = "{Stein}'s Method and Descents after Riffle Shuffles", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "26:901--26:924", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-268", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/268", abstract = "Berestycki and Durrett used techniques from random graph theory to prove that the distance to the identity after iterating the random transposition shuffle undergoes a transition from Poisson to normal behavior. This paper establishes an analogous result for distance after iterates of riffle shuffles or iterates of riffle shuffles and cuts. The analysis uses different tools: Stein's method and generating functions. A useful technique which emerges is that of making a problem more tractable by adding extra symmetry, then using Stein's method to exploit the symmetry in the modified problem, and from this deducing information about the original problem.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Csaki:2005:IPV, author = "Endre Csaki and Yueyun Hu", title = "On the Increments of the Principal Value of {Brownian} Local Time", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "27:925--27:947", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-269", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/269", abstract = "Let $W$ be a one-dimensional Brownian motion starting from 0. Define $ Y(t) = \int_0^t{ds \over W(s)} := \lim_{\epsilon \to 0} \int_0^t 1_{(|W(s)| > \epsilon)} {ds \over W(s)}$ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of $Y$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Chaumont:2005:LPC, author = "Lo{\"\i}c Chaumont and Ronald Doney", title = "On {L{\'e}vy} processes conditioned to stay positive", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "28:948--28:961", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-261", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See corrections \cite{Chaumont:2008:CLP}.", URL = "http://ejp.ejpecp.org/article/view/261", abstract = "We construct the law of L{\'e}vy processes conditioned to stay positive under general hypotheses. We obtain a Williams type path decomposition at the minimum of these processes. This result is then applied to prove the weak convergence of the law of L{\'e}vy processes conditioned to stay positive as their initial state tends to 0. We describe an absolute continuity relationship between the limit law and the measure of the excursions away from 0 of the underlying L{\'e}vy process reflected at its minimum. Then, when the L{\'e}vy process creeps upwards, we study the lower tail at 0 of the law of the height of this excursion.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "L'evy process conditioned to stay positive, path decomposition, weak convergence, excursion measure, creeping", } @Article{Posta:2005:EFO, author = "Gustavo Posta", title = "Equilibrium Fluctuations for a One-Dimensional Interface in the Solid on Solid Approximation", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "29:962--29:987", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-270", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/270", abstract = "An unbounded one-dimensional solid-on-solid model with integer heights is studied. Unbounded here means that there is no {\em a priori} restrictions on the discrete gradient of the interface. The interaction Hamiltonian of the interface is given by a finite range part, proportional to the sum of height differences, plus a part of exponentially decaying long range potentials. The evolution of the interface is a reversible Markov process. We prove that if this system is started in the center of a box of size $L$ after a time of order $ L^3$ it reaches, with a very large probability, the top or the bottom of the box.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Bahlali:2005:GSM, author = "Seid Bahlali and Brahim Mezerdi", title = "A General Stochastic Maximum Principle for Singular Control Problems", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "30:988--30:1004", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-271", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/271", abstract = "We consider the stochastic control problem in which the control domain need not be convex, the control variable has two components, the first being absolutely continuous and the second singular. The coefficients of the state equation are non linear and depend explicitly on the absolutely continuous component of the control. We establish a maximum principle, by using a spike variation on the absolutely continuous part of the control and a convex perturbation on the singular one. This result is a generalization of Peng's maximum principle to singular control problems.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Chorro:2005:CDL, author = "Christophe Chorro", title = "Convergence in {Dirichlet} Law of Certain Stochastic Integrals", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "31:1005--31:1025", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-272", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/272", abstract = "Recently, Nicolas Bouleau has proposed an extension of the Donsker's invariance principle in the framework of Dirichlet forms. He proves that an erroneous random walk of i.i.d random variables converges in Dirichlet law toward the Ornstein--Uhlenbeck error structure on the Wiener space. The aim of this paper is to extend this result to some families of stochastic integrals.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Ganesh:2005:SPL, author = "Ayalvadi Ganesh and Claudio Macci and Giovanni Torrisi", title = "Sample Path Large Deviations Principles for {Poisson} Shot Noise Processes and Applications", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "32:1026--32:1043", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-273", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/273", abstract = "This paper concerns sample path large deviations for Poisson shot noise processes, and applications in queueing theory. We first show that, under an exponential tail condition, Poisson shot noise processes satisfy a sample path large deviations principle with respect to the topology of pointwise convergence. Under a stronger superexponential tail condition, we extend this result to the topology of uniform convergence. We also give applications of this result to determining the most likely path to overflow in a single server queue, and to finding tail asymptotics for the queue lengths at priority queues.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "large deviations; Poisson shot noise; queues; risk; sample paths", } @Article{Bell:2005:DSP, author = "Steven Bell and Ruth Williams", title = "Dynamic Scheduling of a Parallel Server System in Heavy Traffic with Complete Resource Pooling: Asymptotic Optimality of a Threshold Policy", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "33:1044--33:1115", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-281", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/281", abstract = "We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several different classes (or buffers). Jobs within a class are processed on a first-in-first-out basis, where the processing of a given job may be performed by any server from a given (class-dependent) subset of the bank of servers. The random service time of a job may depend on both its class and the server providing the service. Each job departs the system after receiving service from one server. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs to available servers. We consider a parameter regime in which the system satisfies both a heavy traffic and a complete resource pooling condition. Our cost function is an expected cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. In a prior work, the second author proposed a continuous review threshold control policy for use in such a parallel server system. This policy was advanced as an ``interpretation'' of the analytic solution to an associated Brownian control problem (formal heavy traffic diffusion approximation). In this paper we show that the policy proposed previously is asymptotically optimal in the heavy traffic limit and that the limiting cost is the same as the optimal cost in the Brownian control problem.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Ledoux:2005:DIE, author = "Michel Ledoux", title = "Distributions of Invariant Ensembles from the Classical Orthogonal Polynimials: the Discrete Case", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "34:1116--34:1146", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-282", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/282", abstract = "We examine the Charlier, Meixner, Krawtchouk and Hahn discrete orthogonal polynomial ensembles, deeply investigated by K. Johansson, using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. As for the matrix ensembles, equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Factorial moment identities on mean spectral measures may be used towards small deviation inequalities on the rightmost charges at the rate given by the Tracy--Widom asymptotics.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Durrett:2005:CSB, author = "Richard Durrett and Leonid Mytnik and Edwin Perkins", title = "Competing super-{Brownian} motions as limits of interacting particle systems", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "35:1147--35:1220", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-229", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/229", abstract = "We study two-type branching random walks in which the birth or death rate of each type can depend on the number of neighbors of the opposite type. This competing species model contains variants of Durrett's predator-prey model and Durrett and Levin's colicin model as special cases. We verify in some cases convergence of scaling limits of these models to a pair of super-Brownian motions interacting through their collision local times, constructed by Evans and Perkins.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "super-Brownian motion, interacting branching particle systems, collision local time, competing species, measure-valued diffusions", } @Article{Sethuraman:2005:MPD, author = "Sunder Sethuraman and Srinivasa Varadhan", title = "A Martingale Proof of {Dobrushin}'s Theorem for Non-Homogeneous {Markov} Chains", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "36:1221--36:1235", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-283", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/283", abstract = "In 1956, Dobrushin proved an important central limit theorem for non-homogeneous Markov chains. In this note, a shorter and different proof elucidating more the assumptions is given through martingale approximation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Ariyoshi:2005:STA, author = "Teppei Ariyoshi and Masanori Hino", title = "Small-time Asymptotic Estimates in Local {Dirichlet} Spaces", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "37:1236--37:1259", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-286", ISSN = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/286", abstract = "Small-time asymptotic estimates of semigroups on a logarithmic scale are proved for all symmetric local Dirichlet forms on $ \sigma $-finite measure spaces, which is an extension of the work by Hino and Ram{\'\i}rez [4].", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Wang:2005:LTS, author = "Qiying Wang", title = "Limit Theorems for Self-Normalized Large Deviation", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "38:1260--38:1285", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-289", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/289", abstract = "Let $ X, X_1, X_2, \cdots $ be i.i.d. random variables with zero mean and finite variance $ \sigma^2 $. It is well known that a finite exponential moment assumption is necessary to study limit theorems for large deviation for the standardized partial sums. In this paper, limit theorems for large deviation for self-normalized sums are derived only under finite moment conditions. In particular, we show that, if $ E X^4 < \infty $, then \par $$ \frac {P(S_n / V_n \geq x)}{1 - \Phi (x)} = \exp \left \{ - \frac {x^3 EX^3}{3 \sqrt { n} \sigma^3} \right \} \left [1 + O \left (\frac {1 + x}{\sqrt { n}} \right) \right], $$ for $ x \ge 0 $ and $ x = O(n^{1 / 6}) $, where $ S_n = \sum_{i = 1}^n X_i $ and $ V_n = (\sum_{i = 1}^n X_i^2)^{1 / 2} $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Cram{\'e}r large deviation, limit theorem", } @Article{Greven:2005:RTI, author = "Andreas Greven and Vlada Limic and Anita Winter", title = "Representation Theorems for Interacting {Moran} Models, Interacting {Fisher--Wrighter} Diffusions and Applications", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "39:1286--39:1358", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-290", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/290", abstract = "We consider spatially interacting Moran models and their diffusion limit which are interacting Fisher--Wright diffusions. The Moran model is a spatial population model with individuals of different type located on sites given by elements of an Abelian group. The dynamics of the system consists of independent migration of individuals between the sites and a resampling mechanism at each site, i.e., pairs of individuals are replaced by new pairs where each newcomer takes the type of a randomly chosen individual from the parent pair. Interacting Fisher--Wright diffusions collect the relative frequency of a subset of types evaluated for the separate sites in the limit of infinitely many individuals per site. One is interested in the type configuration as well as the time-space evolution of genealogies, encoded in the so-called historical process. The first goal of the paper is the analytical characterization of the historical processes for both models as solutions of well-posed martingale problems and the development of a corresponding duality theory. For that purpose, we link both the historical Fisher--Wright diffusions and the historical Moran models by the so-called look-down process. That is, for any fixed time, a collection of historical Moran models with increasing particle intensity and a particle representation for the limiting historical interacting Fisher--Wright diffusions are provided on one and the same probability space. This leads to a strong form of duality between spatially interacting Moran models, interacting Fisher--Wright diffusions on the one hand and coalescing random walks on the other hand, which extends the classical weak form of moment duality for interacting Fisher--Wright diffusions. Our second goal is to show that this representation can be used to obtain new results on the long-time behavior, in particular (i) on the structure of the equilibria, and of the equilibrium historical processes, and (ii) on the behavior of our models on large but finite site space in comparison with our models on infinite site space. Here the so-called finite system scheme is established for spatially interacting Moran models which implies via the look-down representation also the already known results for interacting Fisher--Wright diffusions. Furthermore suitable versions of the finite system scheme on the level of historical processes are newly developed and verified. In the long run the provided look-down representation is intended to answer questions about finer path properties of interacting Fisher--Wright diffusions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "equilibrium measure; exchangeability; historical martingale problem; historical process; Interacting Fischer--Wright diffusions; large finite systems; look-down construction; spatially interacting Moran model", } @Article{Puchala:2005:EAT, author = "Zbigniew Puchala and Tomasz Rolski", title = "The Exact Asymptotic of the Time to Collision", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "40:1359--40:1380", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-291", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/291", abstract = "In this note we consider the time of the collision $ \tau $ for $n$ independent copies of Markov processes $ X^1_t, \ldots {}, X^n_t$, each starting from $ x_i$, where $ x_1 < \ldots {} < x_n$. We show that for the continuous time random walk $ P_x(\tau > t) = t^{-n(n - 1) / 4}(C h(x) + o(1)), $ where $C$ is known and $ h(x)$ is the Vandermonde determinant. From the proof one can see that the result also holds for $ X_t$ being the Brownian motion or the Poisson process. An application to skew standard Young tableaux is given.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion; collision time; continuous time random walk; skew Young tableaux; tandem queue", } @Article{Igloi:2005:ROT, author = "Endre Igl{\'o}i", title = "A Rate-Optimal Trigonometric Series Expansion of the Fractional {Brownian} Motion", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "41:1381--41:1397", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-287", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/287", abstract = "Let $ B^{(H)}(t), t \in \lbrack - 1, 1] $, be the fractional Brownian motion with Hurst parameter $ H \in (1 / 2, 1) $. In this paper we present the series representation $ B^{(H)}(t) = a_0 t \xi_0 + \sum_{j = 1}^{\infty }a_j((1 - \cos (j \pi t)) \xi_j + \sin (j \pi t) \widetilde {\xi }_j), t \in \lbrack - 1, 1] $, where $ a_j, j \in \mathbb {N} \cup {0} $, are constants given explicitly, and $ \xi_j, j \in \mathbb {N} \cup {0} $, $ \widetilde {\xi }_j, j \in \mathbb {N} $, are independent standard Gaussian random variables. We show that the series converges almost surely in $ C[ - 1, 1] $, and in mean-square (in $ L^2 (\Omega)$), uniformly in $ t \in \lbrack - 1, 1]$. Moreover we prove that the series expansion has an optimal rate of convergence.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "fractional Brownian motion; function series expansion; Gamma-mixed Ornstein--Uhlenbeck process; rate of convergence", } @Article{Mikulevicius:2005:CDP, author = "Remigijus Mikulevicius and Henrikas Pragarauskas", title = "On {Cauchy--Dirichlet} Problem in Half-Space for Linear Integro-Differential Equations in Weighted {H{\"o}lder} Spaces", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "42:1398--42:1416", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-292", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/292", abstract = "We study the Cauchy--Dirichlet problem in half-space for linear parabolic integro-differential equations. Sufficient conditions are derived under which the problem has a unique solution in weighted Hoelder classes. The result can be used in the regularity analysis of certain functionals arising in the theory of Markov processes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov jump processes, parabolic integro-differential equations", } @Article{Jean:2005:RWG, author = "Mairesse Jean", title = "Random Walks on Groups and Monoids with a {Markovian} Harmonic Measure", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "43:1417--43:1441", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-293", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/293", abstract = "We consider a transient nearest neighbor random walk on a group $G$ with finite set of generators $S$. The pair $ (G, S)$ is assumed to admit a natural notion of normal form words where only the last letter is modified by multiplication by a generator. The basic examples are the free products of a finitely generated free group and a finite family of finite groups, with natural generators. We prove that the harmonic measure is Markovian of a particular type. The transition matrix is entirely determined by the initial distribution which is itself the unique solution of a finite set of polynomial equations of degree two. This enables to efficiently compute the drift, the entropy, the probability of ever hitting an element, and the minimal positive harmonic functions of the walk. The results extend to monoids.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Finitely generated group or monoid; free product; harmonic measure.; random walk", } @Article{Kozdron:2005:ERW, author = "Michael Kozdron and Gregory Lawler", title = "Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "44:1442--44:1467", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-294", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/294", abstract = "We prove an estimate for the probability that a simple random walk in a simply connected subset $A$ of $ Z^2$ starting on the boundary exits $A$ at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Cvitanic:2005:SDM, author = "Jaksa Cvitanic and Jianfeng Zhang", title = "The Steepest Descent Method for Forward--Backward {SDEs}", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "45:1468--45:1495", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-295", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/295", abstract = "This paper aims to open a door to Monte-Carlo methods for numerically solving Forward--Backward SDEs, without computing over all Cartesian grids as usually done in the literature. We transform the FBSDE to a control problem and propose the steepest descent method to solve the latter one. We show that the original (coupled) FBSDE can be approximated by {it decoupled} FBSDEs, which further comes down to computing a sequence of conditional expectations. The rate of convergence is obtained, and the key to its proof is a new well-posedness result for FBSDEs. However, the approximating decoupled FBSDEs are non-Markovian. Some Markovian type of modification is needed in order to make the algorithm efficiently implementable.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Hausenblas:2005:EUR, author = "Erika Hausenblas", title = "Existence, Uniqueness and Regularity of Parabolic {SPDEs} Driven by {Poisson} Random Measure", journal = j-ELECTRON-J-PROBAB, volume = "10", pages = "46:1496--46:1546", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v10-297", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/297", abstract = "In this paper we investigate SPDEs in certain Banach spaces driven by a Poisson random measure. We show existence and uniqueness of the solution, investigate certain integrability properties and verify the c{\`a}dl{\`a}g property.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Goel:2006:MTB, author = "Sharad Goel and Ravi Montenegro and Prasad Tetali", title = "Mixing Time Bounds via the Spectral Profile", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "1:1--1:26", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-300", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/300", abstract = "On complete, non-compact manifolds and infinite graphs, Faber--Krahn inequalities have been used to estimate the rate of decay of the heat kernel. We develop this technique in the setting of finite Markov chains, proving upper and lower $ L^{\infty } $ mixing time bounds via the spectral profile. This approach lets us recover and refine previous conductance-based bounds of mixing time (including the Morris--Peres result), and in general leads to sharper estimates of convergence rates. We apply this method to several models including groups with moderate growth, the fractal-like Viscek graphs, and the product group $ Z_a \times Z_b $, to obtain tight bounds on the corresponding mixing times.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Alsmeyer:2006:SFP, author = "Gerold Alsmeyer and Uwe R{\"o}sler", title = "A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "2:27--2:56", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-296", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/296", abstract = "For real numbers $ C, T_1, T_2, \ldots {} $ we find all solutions $ \mu $ to the stochastic fixed point equation $ W \sim \sum_{j \ge 1}T_j W_j + C $, where $ W, W_1, W_2, \ldots {} $ are independent real-valued random variables with distribution $ \mu $ and $ \sim $ means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of $ { R}_*= { R} \backslash \{ 0 \} $ generated by the $ T_j $. If this group is continuous, i.e., $ {R}_* $ itself or the positive half line $ {R}_+ $, then all nontrivial fixed points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Levy measure of any fixed point is harmonic with respect to $ \Lambda = \sum_{j \ge 1} \delta_{T_j} $, i.e., $ \Gamma = \Gamma \star \Lambda $, where $ \star $ means multiplicative convolution. This will enable us to apply the powerful Choquet--Deny theorem.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Choquet--Deny theorem; infinite divisibility; L'evy measure; stable distribution; Stochastic fixed point equation; weighted branching process", } @Article{Cheridito:2006:DMR, author = "Patrick Cheridito and Freddy Delbaen and Michael Kupper", title = "Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "3:57--3:106", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-302", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/302", abstract = "We study dynamic monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a dynamic risk measure time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time. We show that this condition translates into a decomposition property for the corresponding acceptance sets, and we demonstrate how time-consistent dynamic monetary risk measures can be constructed by pasting together one-period risk measures. For conditional coherent and convex monetary risk measures, we provide dual representations of Legendre--Fenchel type based on linear functionals induced by adapted increasing processes of integrable variation. Then we give dual characterizations of time-consistency for dynamic coherent and convex monetary risk measures. To this end, we introduce a concatenation operation for adapted increasing processes of integrable variation, which generalizes the pasting of probability measures. In the coherent case, time-consistency corresponds to stability under concatenation in the dual. For dynamic convex monetary risk measures, the dual characterization of time-consistency generalizes to a condition on the family of convex conjugates of the conditional risk measures at different times. The theoretical results are applied by discussing the time-consistency of various specific examples of dynamic monetary risk measures that depend on bounded discrete-time processes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Tang:2006:IND, author = "Qihe Tang", title = "Insensitivity to Negative Dependence of the Asymptotic Behavior of Precise Large Deviations", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "4:107--4:120", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-304", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/304", abstract = "Since the pioneering works of C. C. Heyde, A. V. Nagaev, and S. V. Nagaev in 1960's and 1970's, the precise asymptotic behavior of large-deviation probabilities of sums of heavy-tailed random variables has been extensively investigated by many people, but mostly it is assumed that the random variables under discussion are independent. In this paper, we extend the study to the case of negatively dependent random variables and we find out that the asymptotic behavior of precise large deviations is insensitive to the negative dependence.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "(lower/upper) negative dependence; (upper) Matuszewska index; Consistent variation; partial sum; precise large deviations; uniform asymptotics", } @Article{Hamadene:2006:BTR, author = "Said Hamadene and Mohammed Hassani", title = "{BSDEs} with two reflecting barriers driven by a {Brownian} motion and {Poisson} noise and related {Dynkin} game", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "5:121--5:145", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-303", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/303", abstract = "In this paper we study BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson process. We show the existence and uniqueness of {\em local\/} and global solutions. As an application we solve the related zero-sum Dynkin game.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Backward stochastic differential equation; Dynkin game; Mokobodzki's condition; Poisson measure", } @Article{Song:2006:TSE, author = "Renming Song", title = "Two-sided Estimates on the Density of the {Feynman--Kac} Semigroups of Stable-like Processes", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "6:146--6:161", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-308", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/308", abstract = "In this paper we establish two-sided estimates for the density of the Feynman--Kac semigroups of stable-like processes with potentials given by signed measures belonging to the Kato class. We also provide similar estimates for the densities of two other kinds of Feynman--Kac semigroups of stable-like processes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "continuous additive functionals; continuous additive functionals of zero energy; Feynman--Kac semigroups; Kato class; purely discontinuous additive functionals.; Stable processes; stable-like processes", } @Article{Tsirelson:2006:BLM, author = "Boris Tsirelson", title = "{Brownian} local minima, random dense countable sets and random equivalence classes", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "7:162--7:198", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-309", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/309", abstract = "A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts, proposed here, includes a wide class of random equivalence classes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion; equivalence relation; local minimum; point process", } @Article{Picard:2006:BES, author = "Jean Picard", title = "{Brownian} excursions, stochastic integrals, and representation of {Wiener} functionals", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "8:199--8:248", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-310", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/310", abstract = "A stochastic calculus similar to Malliavin's calculus is worked out for Brownian excursions. The analogue of the Malliavin derivative in this calculus is not a differential operator, but its adjoint is (like the Skorohod integral) an extension of the It{\^o} integral. As an application, we obtain an expression for the integrand in the stochastic integral representation of square integrable Wiener functionals; this expression is an alternative to the classical Clark--Ocone formula. Moreover, this calculus enables to construct stochastic integrals of predictable or anticipating processes (forward, backward and symmetric integrals are considered).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "anticipating calculus; Brownian excursions; Malliavin calculus; stochastic integral representation; stochastic integrals", } @Article{Etore:2006:RWS, author = "Pierre Etor{\'e}", title = "On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "9:249--9:275", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-311", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/311", abstract = "In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coefficients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Monte Carlo methods, random walk, Skew Brownian motion, one-dimensional process, divergence form operator", } @Article{Bavouzet:2006:CGU, author = "Marie Pierre Bavouzet and Marouen Messaoud", title = "Computation of {Greeks} using {Malliavin}'s calculus in jump type market models", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "10:276--10:300", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-314", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/314", abstract = "We use the Malliavin calculus for Poisson processes in order to compute sensitivities for European and Asian options with underlying following a jump type diffusion. The main point is to settle an integration by parts formula (similar to the one in the Malliavin calculus) for a general multidimensional random variable which has an absolutely continuous law with differentiable density. We give an explicit expression of the differential operators involved in this formula and this permits to simulate them and consequently to run a Monte Carlo algorithm", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Asian options; compound Poisson process; Euler scheme; European options; Malliavin calculus; Monte-Carlo algorithm; sensitivity analysis", } @Article{Sellke:2006:RRR, author = "Thomas Sellke", title = "Recurrence of Reinforced Random Walk on a Ladder", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "11:301--11:310", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-313", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/313", abstract = "Consider reinforced random walk on a graph that looks like a doubly infinite ladder. All edges have initial weight 1, and the reinforcement convention is to add $ \delta > 0 $ to the weight of an edge upon first crossing, with no reinforcement thereafter. This paper proves recurrence for all $ \delta > 0 $. In so doing, we introduce a more general class of processes, termed multiple-level reinforced random walks.\par {\bf Editor's Note}. A draft of this paper was written in 1994. The paper is one of the first to make any progress on this type of reinforcement problem. It has motivated a substantial number of new and sometimes quite difficult studies of reinforcement models in pure and applied probability. The persistence of interest in models related to this has caused the original unpublished manuscript to be frequently cited, despite its lack of availability and the presence of errors. The opportunity to rectify this situation has led us to the somewhat unusual step of publishing a result that may have already entered the mathematical folklore.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "learning; Markov; martingale; multiple-level; Reinforced Random Walk", } @Article{Grigorescu:2006:TPL, author = "Ilie Grigorescu and Min Kang", title = "Tagged Particle Limit for a {Fleming--Viot} Type System", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "12:311--12:331", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-316", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/316", abstract = "We consider a branching system of $N$ Brownian particles evolving independently in a domain $D$ during any time interval between boundary hits. As soon as one particle reaches the boundary it is killed and one of the other particles splits into two independent particles, the complement of the set $D$ acting as a catalyst or hard obstacle. Identifying the newly born particle with the one killed upon contact with the catalyst, we determine the exact law of the tagged particle as $N$ approaches infinity. In addition, we show that any finite number of labelled particles become independent in the limit. Both results can be seen as scaling limits of a genome population undergoing redistribution present in the Fleming--Viot dynamics.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Fleming--Viot, propagation of chaos, tagged particle", } @Article{Deijfen:2006:NCR, author = "Maria Deijfen and Olle H{\"a}ggstr{\"o}m", title = "Nonmonotonic Coexistence Regions for the Two-Type {Richardson} Model on Graphs", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "13:331--13:344", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-321", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/321", abstract = "In the two-type Richardson model on a graph $ G = (V, E) $, each vertex is at a given time in state $0$, $1$ or $2$. A $0$ flips to a $1$ (resp.\ $2$) at rate $ \lambda_1$ ($ \lambda_2$) times the number of neighboring $1$'s ($2$'s), while $1$'s and $2$'s never flip. When $G$ is infinite, the main question is whether, starting from a single $1$ and a single $2$, with positive probability we will see both types of infection reach infinitely many sites. This has previously been studied on the $d$-dimensional cubic lattice $ Z^d$, $ d \geq 2$, where the conjecture (on which a good deal of progress has been made) is that such coexistence has positive probability if and only if $ \lambda_1 = \lambda_2$. In the present paper examples are given of other graphs where the set of points in the parameter space which admit such coexistence has a more surprising form. In particular, there exist graphs exhibiting coexistence at some value of $ \frac {\lambda_1}{\lambda_2} \neq 1$ and non-coexistence when this ratio is brought closer to $1$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "coexistence; Competing growth; graphs", } @Article{Caravenna:2006:SAB, author = "Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti", title = "Sharp asymptotic behavior for wetting models in (1+1)-dimension", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "14:345--14:362", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-320", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/320", abstract = "We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure in all regimes, including the critical one.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Critical Wetting; delta-Pinning Model; Fluctuation Theory for Random Walks; Renewal Theory; Wetting Transition", } @Article{Limic:2006:SC, author = "Vlada Limic and Anja Sturm", title = "The spatial {$ \Lambda $}-coalescent", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "15:363--15:393", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-319", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/319", abstract = "This paper extends the notion of the $ \Lambda $-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $ \Lambda $-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $ \Lambda $-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $ \Lambda $-coalescents on large tori in $ d \geq 3$ dimensions. Some of our results generalize and strengthen the corresponding results in Greven et al. (2005) concerning the spatial Kingman coalescent.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "$la$-coalescent; coalescent; limit theorems, coalescing random walks; structured coalescent", } @Article{Basdevant:2006:FOP, author = "Anne-Laure Basdevant", title = "Fragmentation of Ordered Partitions and Intervals", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "16:394--16:417", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-323", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/323", abstract = "Fragmentation processes of exchangeable partitions have already been studied by several authors. This paper deals with fragmentations of exchangeable compositions, i.e., partitions of $ \mathbb {N} $ in which the order of the blocks matters. We will prove that such a fragmentation is bijectively associated to an interval fragmentation. Using this correspondence, we then study two examples: Ruelle's interval fragmentation and the interval fragmentation derived from the standard additive coalescent.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "exchangeable compositions; Interval fragmentation", } @Article{Holroyd:2006:MTM, author = "Alexander Holroyd", title = "The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "17:418--17:433", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-326", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/326", abstract = "In the modified bootstrap percolation model, sites in the cube $ \{ 1, \ldots, L \}^d $ are initially declared active independently with probability $p$. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the $d$ dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all $ d \geq 2$ we prove that as $ L \to \infty $ and $ p \to 0$ simultaneously, this probability converges to $1$ if $ L \geq \exp \cdots \exp \frac {\lambda + \epsilon }{p}$, and converges to $0$ if $ L \leq \exp \cdots \exp \frac {\lambda - \epsilon }{p}$, for any $ \epsilon > 0$. Here the exponential function is iterated $ d - 1$ times, and the threshold $ \lambda $ equals $ \pi^2 / 6$ for all $d$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "bootstrap percolation; cellular automaton; finite-size scaling; metastability", } @Article{Nane:2006:LIL, author = "Erkan Nane", title = "Laws of the iterated logarithm for $ \alpha $-time {Brownian} motion", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "18:434--18:459", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-327", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/327", abstract = "We introduce a class of iterated processes called $ \alpha $-time Brownian motion for $ 0 < \alpha \leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $ \alpha $-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in {citehu} for iterated Brownian motion. When $ \alpha = 1$ it takes the following form\par $$ \liminf_{T \to \infty } \ T^{-1 / 2}(\log \log T) \sup_{0 \leq t \leq T}|Z_t| = \pi^2 \sqrt {\lambda_1} \quad a.s. $$ where $ \lambda_1$ is the first eigenvalue for the Cauchy process in the interval $ [ - 1, 1].$ We also define the local time $ L^*(x, t)$ and range $ R^*(t) = |{x \colon Z(s) = x \text { for some } s \leq t}|$ for these processes for $ 1 < \alpha < 2$. We prove that there are universal constants $ c_R, c_L \in (0, \infty) $ such that\par $$ \limsup_{t \to \infty } \frac {R^*(t)}{(t / \log \log t)^{1 / 2 \alpha } \log \log t} = c_R \quad a.s. $$ $$ \liminf_{t \to \infty } \frac {\sup_{x \in {R}}L^*(x, t)}{(t / \log \log t)^{1 - 1 / 2 \alpha }} = c_L \quad a.s. $$", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, symmetric $alpha$-stable process, $alpha$-time Brownian motion, local time, Chung's law, Kesten's law", } @Article{Adams:2006:LSP, author = "Stefan Adams and Jean-Bernard Bru and Wolfgang Koenig", title = "Large systems of path-repellent {Brownian} motions in a trap at positive temperature", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "19:460--19:485", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-330", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/330", abstract = "We study a model of $N$ mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the motions from escaping to infinity, and a pair-interaction Hamiltonian, which imposes a repellency of the $N$ paths. In fact, this interaction is an $N$-dependent regularisation of the Brownian intersection local times, an object which is of independent interest in the theory of stochastic processes. The time horizon (interpreted as the inverse temperature) is kept fixed. We analyse the model for diverging number of Brownian motions in terms of a large deviation principle. The resulting variational formula is the positive-temperature analogue of the well-known Gross--Pitaevskii formula, which approximates the ground state of a certain dilute large quantum system; the kinetic energy term of that formula is replaced by a probabilistic energy functional. This study is a continuation of the analysis in [ABK06] where we considered the limit of diverging time (i.e., the zero-temperature limit) with fixed number of Brownian motions, followed by the limit for diverging number of motions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian intersection local times; Gross--Pitaevskii formula; Interacting Brownian motions; large deviations; occupation measure", } @Article{Klein:2006:CCI, author = "Thierry Klein and Yutao Ma and Nicolas Privault", title = "Convex Concentration Inequalities and Forward--Backward Stochastic Calculus", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "20:486--20:512", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-332", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/332", abstract = "Given $ (M_t)_{t \in \mathbb {R}_+} $ and $ (M^*_t)_{t \in \mathbb {R}_+} $ respectively a forward and a backward martingale with jumps and continuous parts, we prove that $ E[\phi (M_t + M^*_t)] $ is non-increasing in $t$ when $ \phi $ is a convex function, provided the local characteristics of $ (M_t)_{t \in \mathbb {R}_+}$ and $ (M^*_t)_{t \in \mathbb {R}_+}$ satisfy some comparison inequalities. We deduce convex concentration inequalities and deviation bounds for random variables admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Convex concentration inequalities, forward--backward stochastic calculus, deviation inequalities, Clark formula, Brownian motion, jump processes", } @Article{Maximilian:2006:EMD, author = "Duerre Maximilian", title = "Existence of multi-dimensional infinite volume self-organized critical forest-fire models", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "21:513--21:539", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-333", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/333", abstract = "Consider the following forest-fire model where the possible locations of trees are the sites of a cubic lattice. Each site has two possible states: 'vacant' or 'occupied'. Vacant sites become occupied according to independent rate 1 Poisson processes. Independently, at each site ignition (by lightning) occurs according to independent rate lambda Poisson processes. When a site is ignited, its occupied cluster becomes vacant instantaneously. If the lattice is one-dimensional or finite, then with probability one, at each time the state of a given site only depends on finitely many Poisson events; a process with the above description can be constructed in a standard way. If the lattice is infinite and multi-dimensional, in principle, the state of a given site can be influenced by infinitely many Poisson events in finite time.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "existence; forest-fire model; forest-fires; self-organized criticality; well-defined", } @Article{Schmitz:2006:ECD, author = "Tom Schmitz", title = "Examples of Condition {$ (T) $} for Diffusions in a Random Environment", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "22:540--22:562", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-337", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/337", abstract = "With the help of the methods developed in our previous article [Schmitz, to appear in Annales de l'I.H.P., in press], we highlight condition $ (T) $ as a source of new examples of 'ballistic' diffusions in a random environment when $ d > 1 $ ('ballistic' means that a strong law of large numbers with non-vanishing limiting velocity holds). In particular we are able to treat the case of non-constant diffusion coefficients, a feature that causes problems. Further we recover the ballistic character of two important classes of diffusions in a random environment by simply checking condition $ (T) $. This not only points out to the broad range of examples where condition $ (T) $ can be checked, but also fortifies our belief that condition $ (T) $ is a natural contender for the characterisation of ballistic diffusions in a random environment when $ d > 1 $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Diffusions in a random environment, ballistic behavior, Condition $(T)$", } @Article{Kim:2006:PSD, author = "Kyeong-Hun Kim", title = "Parabolic {SPDEs} Degenerating on the Boundary of Non-Smooth Domain", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "23:563--23:584", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-339", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/339", abstract = "Degenerate stochastic partial differential equations of divergence and non-divergence forms are considered in non-smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces, and Holder estimates of the solutions are presented.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "SPDEs degenerating on the boundary; weighted Sobolev spaces", } @Article{Swart:2006:RAC, author = "Jan Swart and Klaus Fleischmann", title = "Renormalization analysis of catalytic {Wright--Fisher} diffusions", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "24:585--24:654", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-341", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/341", abstract = "Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions indexed by the hierarchical group, with a linear attractive interaction between the components. In this context, the mappings are called renormalization transformations. We consider such maps for catalytic Wright--Fisher diffusions. These are diffusions on the unit square where the first component (the catalyst) performs an autonomous Wright--Fisher diffusion, while the second component (the reactant) performs a Wright--Fisher diffusion with a rate depending on the first component through a catalyzing function. We determine the limit of rescaled iterates of renormalization transformations acting on the diffusion matrices of such catalytic Wright--Fisher diffusions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Renormalization, catalytic Wright--Fisher diffusion, embedded particle system, extinction, unbounded growth, interacting diffusions, universality", } @Article{Berger:2006:TPC, author = "Noam Berger and Itai Benjamini and Omer Angel and Yuval Peres", title = "Transience of percolation clusters on wedges", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "25:655--25:669", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-345", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/345", abstract = "We study random walks on supercritical percolation clusters on wedges in $ Z^3 $, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. H{\"a}ggstr{\"o}m and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on $ Z^2 $ is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This answers a question of C. Hoffman.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "percolation; transience; wedges", } @Article{Cator:2006:BSC, author = "Eric Cator and Sergei Dobrynin", title = "Behavior of a second class particle in {Hammersley}'s process", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "26:670--26:685", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-340", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/340", abstract = "In the case of a rarefaction fan in a non-stationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we consider a stationary Hammersley process and use the previous results to show that trajectories of a second class particle and a dual second class particles touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. This is linked to the area of influence of an added Poisson point in the plane.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Hammersley's process; rarefaction fan; second class particles", } @Article{Odasso:2006:SSS, author = "Cyril Odasso", title = "Spatial Smoothness of the Stationary Solutions of the {$3$D} {Navier--Stokes} Equations", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "27:686--27:699", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-336", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/336", abstract = "We consider stationary solutions of the three dimensional Navier--Stokes equations (NS3D) with periodic boundary conditions and driven by an external force which might have a deterministic and a random part. The random part of the force is white in time and very smooth in space. We investigate smoothness properties in space of the stationary solutions. Classical technics for studying smoothness of stochastic PDEs do not seem to apply since global existence of strong solutions is not known. We use the Kolmogorov operator and Galerkin approximations. We first assume that the noise has spatial regularity of order $p$ in the $ L^2$ based Sobolev spaces, in other words that its paths are in $ H^p$. Then we prove that at each fixed time the law of the stationary solutions is supported by $ H^{p + 1}$. Then, using a totally different technic, we prove that if the noise has Gevrey regularity then at each fixed time, the law of a stationary solution is supported by a Gevrey space. Some information on the Kolmogorov dissipation scale is deduced.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic three-dimensional Navier--Stokes equations, invariant measure", } @Article{Dereich:2006:HRQ, author = "Steffen Dereich and Michael Scheutzow", title = "High Resolution Quantization and Entropy Coding for Fractional {Brownian} Motion", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "28:700--28:722", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-344", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/344", abstract = "We establish the precise asymptotics of the quantization and entropy coding errors for fractional Brownian motion with respect to the supremum norm and $ L^p [0, 1]$-norm distortions. We show that all moments in the quantization problem lead to the same asymptotics. Using a general principle, we conclude that entropy coding and quantization coincide asymptotically. Under supremum-norm distortion, our proof uses an explicit construction of efficient codebooks based on a particular entropy constrained coding scheme.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "complexity; distortion rate function; entropy; High-resolution quantization; stochastic process", } @Article{Fleischmann:2006:HLF, author = "Klaus Fleischmann and Peter M{\"o}rters and Vitali Wachtel", title = "Hydrodynamic Limit Fluctuations of Super-{Brownian} Motion with a Stable Catalyst", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "29:723--29:767", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-348", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/348", abstract = "We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein--Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a Gaussian situation to stable fluctuations of index $ 1 + \gamma $, where $ \gamma \in (0, 1) $ is an index associated to the medium.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Catalyst, reactant, superprocess, critical scaling, refined law of large numbers, catalytic branching, stable medium, random environment, supercritical dimension, generalised stable Ornstein--Uhlenbeck process, index jump, parabolic Anderson model with sta", } @Article{Belhaouari:2006:CRS, author = "Samir Belhaouari and Thomas Mountford and Rongfeng Sun and Glauco Valle", title = "Convergence Results and Sharp Estimates for the Voter Model Interfaces", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "30:768--30:801", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-349", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/349", abstract = "We study the evolution of the interface for the one-dimensional voter model. We show that if the random walk kernel associated with the voter model has finite $ \gamma $-th moment for some $ \gamma > 3$, then the evolution of the interface boundaries converge weakly to a Brownian motion under diffusive scaling. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that finite $ \gamma $-th moment is necessary for this convergence for all $ \gamma \in (0, 3)$. We also obtain relatively sharp estimates for the tail distribution of the size of the equilibrium interface, extending earlier results of Cox and Durrett, and Belhaouari, Mountford and Valle.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "voter model interface, coalescing random walks, Brownian web, invariance principle", } @Article{Sabot:2006:RWD, author = "Christophe Sabot and Nathana{\"e}l Enriquez", title = "Random Walks in a {Dirichlet} Environment", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "31:802--31:816", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-350", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/350", abstract = "This paper states a law of large numbers for a random walk in a random iid environment on $ Z^d $, where the environment follows some Dirichlet distribution. Moreover, we give explicit bounds for the asymptotic velocity of the process and also an asymptotic expansion of this velocity at low disorder.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random Walks, Random Environments, Dirichlet Laws, Reinforced Random Walks", } @Article{Xiao:2006:SLN, author = "Yimin Xiao and Davar Khoshnevisan and Dongsheng Wu", title = "Sectorial Local Non-Determinism and the Geometry of the {Brownian} Sheet", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "32:817--32:843", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-353", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/353", abstract = "We prove the following results about the images and multiple points of an $N$-parameter, $d$-dimensional Brownian sheet $ B = \{ B(t) \}_{t \in R_+^N}$: (1) If $ \text {dim}_H F \leq d / 2$, then $ B(F)$ is almost surely a Salem set.\par (2) If $ N \leq d / 2$, then with probability one $ \text {dim}_H B(F) = 2 \text {dim} F$ for all Borel sets of $ R_+^N$, where ``$ \text {dim}_H$'' could be everywhere replaced by the ``Hausdorff, '' ``packing, '' ``upper Minkowski, '' or ``lower Minkowski dimension.''\par (3) Let $ M_k$ be the set of $k$-multiple points of $B$. If $ N \leq d / 2$ and $ N k > (k - 1)d / 2$, then $ \text {dim}_H M_k = \text {dim}_p M_k = 2 N k - (k - 1)d$, a.s.\par The Hausdorff dimension aspect of (2) was proved earlier; see Mountford (1989) and Lin (1999). The latter references use two different methods; ours of (2) are more elementary, and reminiscent of the earlier arguments of Monrad and Pitt (1987) that were designed for studying fractional Brownian motion. If $ N > d / 2$ then (2) fails to hold. In that case, we establish uniform-dimensional properties for the $ (N, 1)$-Brownian sheet that extend the results of Kaufman (1989) for 1-dimensional Brownian motion. Our innovation is in our use of the {\em sectorial local nondeterminism} of the Brownian sheet (Khoshnevisan and Xiao, 2004).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian sheet, sectorial local nondeterminism, image, Salem sets, multiple points, Hausdorff dimension, packing dimension", } @Article{Dony:2006:WUC, author = "Julia Dony and Uwe Einmahl", title = "Weighted uniform consistency of kernel density estimators with general bandwidth sequences", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "33:844--33:859", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-354", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/354", abstract = "Let $ f_{n, h} $ be a kernel density estimator of a continuous and bounded $d$-dimensional density $f$. Let $ \psi (t)$ be a positive continuous function such that $ \| \psi f^\beta \|_\infty < \infty $ for some $ 0 < \beta < 1 / 2$. We are interested in the rate of consistency of such estimators with respect to the weighted sup-norm determined by $ \psi $. This problem has been considered by Gin, Koltchinskii and Zinn (2004) for a deterministic bandwidth $ h_n$. We provide ``uniform in $h$'' versions of some of their results, allowing us to determine the corresponding rates of consistency for kernel density estimators where the bandwidth sequences may depend on the data and/or the location.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "convergence rates; empirical process; kernel density estimator; uniform in bandwidth; weighted uniform consistency", } @Article{Feyel:2006:CIA, author = "Denis Feyel and Arnaud {de La Pradelle}", title = "Curvilinear Integrals Along Enriched Paths", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "34:860--34:892", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-356", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/356", abstract = "Inspired by the fundamental work of T. J. Lyons, we develop a theory of curvilinear integrals along a new kind of enriched paths in $ R^d $. We apply these methods to the fractional Brownian Motion, and prove a support theorem for SDE driven by the Skorohod fBM of Hurst parameter $ H > 1 / 4 $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Curvilinear integrals, H{\"o}lder continuity, rough paths, stochastic integrals, stochastic differential equations, fractional Brownian motion.", } @Article{Wagner:2006:PGB, author = "Wolfgang Wagner", title = "Post-gelation behavior of a spatial coagulation model", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "35:893--35:933", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-359", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/359", abstract = "A coagulation model on a finite spatial grid is considered. Particles of discrete masses jump randomly between sites and, while located at the same site, stick together according to some coagulation kernel. The asymptotic behavior (for increasing particle numbers) of this model is studied in the situation when the coagulation kernel grows sufficiently fast so that the phenomenon of gelation is observed. Weak accumulation points of an appropriate sequence of measure-valued processes are characterized in terms of solutions of a nonlinear equation. A natural description of the behavior of the gel is obtained by using the one-point compactification of the size space. Two aspects of the limiting equation are of special interest. First, for a certain class of coagulation kernels, this equation differs from a naive extension of Smoluchowski's coagulation equation. Second, due to spatial inhomogeneity, an equation for the time evolution of the gel mass density has to be added. The jump rates are assumed to vanish with increasing particle masses so that the gel is immobile. Two different gel growth mechanisms (active and passive gel) are found depending on the type of the coagulation kernel.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "post-gelation behavior; Spatial coagulation model; stochastic particle systems", } @Article{Ramanan:2006:RDD, author = "Kavita Ramanan", title = "Reflected Diffusions Defined via the Extended {Skorokhod} Map", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "36:934--36:992", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-360", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/360", abstract = "This work introduces the extended Skorokhod problem (ESP) and associated extended Skorokhod map (ESM) that enable a pathwise construction of reflected diffusions that are not necessarily semimartingales. Roughly speaking, given the closure $G$ of an open connected set in $ {\mathbb R}^J$, a non-empty convex cone $ d(x) \subset {\mathbb R}^J$ specified at each point $x$ on the boundary $ \partial G$, and a c{\`a}dl{\`a}g trajectory $ \psi $ taking values in $ {\mathbb R}^J$, the ESM $ \bar \Gamma $ defines a constrained version $ \phi $ of $ \psi $ that takes values in $G$ and is such that the increments of $ \phi - \psi $ on any interval $ [s, t]$ lie in the closed convex hull of the directions $ d(\phi (u)), u \in (s, t]$. When the graph of $ d(\cdot)$ is closed, the following three properties are established: (i) given $ \psi $, if $ (\phi, \eta)$ solve the ESP then $ (\phi, \eta)$ solve the corresponding Skorokhod problem (SP) if and only if $ \eta $ is of bounded variation; (ii) given $ \psi $, any solution $ (\phi, \eta)$ to the ESP is a solution to the SP on the interval $ [0, \tau_0)$, but not in general on $ [0, \tau_0]$, where $ \tau_0$ is the first time that $ \phi $ hits the set $ {\cal V}$ of points $ x \in \partial G$ such that $ d(x)$ contains a line; (iii) the graph of the ESM $ \bar \Gamma $ is closed on the space of c{\`a}dl{\`a}g trajectories (with respect to both the uniform and the $ J_1$-Skorokhod topologies).\par The paper then focuses on a class of multi-dimensional ESPs on polyhedral domains with a non-empty $ {\cal V}$-set. Uniqueness and existence of solutions for this class of ESPs is established and existence and pathwise uniqueness of strong solutions to the associated stochastic differential equations with reflection is derived. The associated reflected diffusions are also shown to satisfy the corresponding submartingale problem. Lastly, it is proved that these reflected diffusions are semimartingales on $ [0, \tau_0]$. One motivation for the study of this class of reflected diffusions is that they arise as approximations of queueing networks in heavy traffic that use the so-called generalised processor sharing discipline.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "reflected diffusions; Skorokhod problem; stochastic differential equations; submartingale problem", } @Article{Bass:2006:MDL, author = "Richard Bass and Xia Chen and Jay Rosen", title = "Moderate deviations and laws of the iterated logarithm for the renormalized self-intersection local times of planar random walks", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "37:993--37:1030", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-362", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/362", abstract = "We study moderate deviations for the renormalized self-intersection local time of planar random walks. We also prove laws of the iterated logarithm for such local times.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion; Gagliardo--Nirenberg; intersection local time; large deviations; law of the iterated logarithm; moderate deviations; planar random walks", } @Article{Gapeev:2006:DOS, author = "Pavel Gapeev", title = "Discounted optimal stopping for maxima in diffusion models with finite horizon", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "38:1031--38:1048", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-367", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/367", abstract = "We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "a change-of-varia; a nonlinear Volterra integral equation of the second kind; boundary surface; Discounted optimal stopping problem; finite horizon; geometric Brownian motion; maximum process; normal reflection; parabolic free-boundary problem; smooth fit", } @Article{Pinelis:2006:NDS, author = "Iosif Pinelis", title = "On normal domination of (super)martingales", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "39:1049--39:1070", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-371", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/371", abstract = "Let $ (S_0, S_1, \dots) $ be a supermartingale relative to a nondecreasing sequence of $ \sigma $-algebras $ (H_{\le 0}, H_{\le 1}, \dots)$, with $ S_0 \leq 0$ almost surely (a.s.) and differences $ X_i := S_i - S_{i - 1}$. Suppose that for every $ i = 1, 2, \dots $ there exist $ H_{\le (i - 1)}$-measurable r.v.'s $ C_{i - 1}$ and $ D_{i - 1}$ and a positive real number $ s_i$ such that $ C_{i - 1} \leq X_i \le D_{i - 1}$ and $ D_{i - 1} - C_{i - 1} \leq 2 s_i$ a.s. Then for all real $t$ and natural $n$ and all functions $f$ satisfying certain convexity conditions $ E f(S_n) \leq E f(s Z)$, where $ f_t(x) := \max (0, x - t)^5$, $ s := \sqrt {s_1^2 + \dots + s_n^2}$, and $ Z \sim N(0, 1)$. In particular, this implies $ P(S_n \ge x) \le c_{5, 0}P(s Z \ge x) \quad \forall x \in R$, where $ c_{5, 0} = 5 !(e / 5)^5 = 5.699 \dots $. Results for $ \max_{0 \leq k \leq n}S_k$ in place of $ S_n$ and for concentration of measure also follow.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "generalized moments; martingales; probability inequalities; supermartingales; upper bounds", } @Article{Chazottes:2006:REW, author = "Jean-Ren{\'e} Chazottes and Cristian Giardina and Frank Redig", title = "Relative entropy and waiting times for continuous-time {Markov} processes", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "40:1049--40:1068", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-374", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/374", abstract = "For discrete-time stochastic processes, there is a close connection between return (resp. waiting) times and entropy (resp. relative entropy). Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one needs a reference measure on path space and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of the logarithm of waiting-times ratios suitably normalized, and their fluctuation properties (central limit theorem and large deviation principle).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "continuous-time Markov chain, law of large numbers, central limit theorem, large deviations, entropy production, time-reversed process", } @Article{Zhan:2006:SPA, author = "Dapeng Zhan", title = "Some Properties of Annulus {SLE}", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "41:1069--41:1093", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-338", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/338", abstract = "An annulus SLE$_\kappa $ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus SLE$_4$, SLE$_8$ and SLE$_8 / 3$. From the local martingale for annulus SLE$_4$ we find a candidate of discrete lattice model that may have annulus SLE$_4$ as its scaling limit. The local martingale for annulus SLE$_8 / 3$ is similar to those for chordal and radial SLE$_8 / 3$. But it seems that annulus SLE$_8 / 3$ does not satisfy the restriction property", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "continuum scaling limit, percolation, SLE, conformal invariance", } @Article{Balazs:2006:CRF, author = "Marton Balazs and Eric Cator and Timo Seppalainen", title = "Cube Root Fluctuations for the Corner Growth Model Associated to the Exclusion Process", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "42:1094--42:1132", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-366", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/366", abstract = "We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order $ t^{2 / 3} $. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order $ t^{1 / 3} $, and also that the transversal fluctuations of the maximal path have order $ t^{2 / 3} $. We adapt and then build on a recent study of Hammersley's process by Cator and Groeneboom, and also utilize the competition interface introduced by Ferrari, Martin and Pimentel. The arguments are entirely probabilistic, and no use is made of the combinatorics of Young tableaux or methods of asymptotic analysis.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Burke's theorem; competition interface; cube root asymptotics; Last-passage; rarefaction fan; simple exclusion", } @Article{Brouwer:2006:CSD, author = "Rachel Brouwer and Juho Pennanen", title = "The Cluster Size Distribution for a Forest-Fire Process on {$Z$}", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "43:1133--43:1143", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-369", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/369", abstract = "Consider the following forest-fire model where trees are located on sites of $ \mathbb {Z} $. A site can be vacant or be occupied by a tree. Each vacant site becomes occupied at rate $1$, independently of the other sites. Each site is hit by lightning with rate $ \lambda $, which burns down the occupied cluster of that site instantaneously. As $ \lambda \downarrow 0$ this process is believed to display self-organised critical behaviour.\par This paper is mainly concerned with the cluster size distribution in steady-state. Drossel, Clar and Schwabl (1993) claimed that the cluster size distribution has a certain power law behaviour which holds for cluster sizes that are not too large compared to some explicit cluster size $ s_{max}$. The latter can be written in terms of $ \lambda $ approximately as $ s_{max} \ln (s_{max}) = 1 / \lambda $. However, Van den Berg and Jarai (2005) showed that this claim is not correct for cluster sizes of order $ s_{max}$, which left the question for which cluster sizes the power law behaviour {\em does} hold. Our main result is a rigorous proof of the power law behaviour up to cluster sizes of the order $ s_{max}^{1 / 3}$. Further, it proves the existence of a stationary translation invariant distribution, which was always assumed but never shown rigorously in the literature.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "forest-fires, self-organised criticality, cluster size distribution", } @Article{Shiga:2006:IDR, author = "Tokuzo Shiga and Hiroshi Tanaka", title = "Infinitely Divisible Random Probability Distributions with an Application to a Random Motion in a Random Environment", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "44:1144--44:1183", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-380", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/380", abstract = "The infinite divisibility of probability distributions on the space $ P (R) $ of probability distributions on $R$ is defined and related fundamental results such as the L{\'e}vy--Khintchin formula, representation of It{\^o} type of infinitely divisible RPD, stable RPD and Levy processes on $ P (R)$ are obtained. As an application we investigate limiting behaviors of a simple model of a particle motion in a random environment", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "infinite divisibility; L{\'e}vy-It{\^o} repr{\'e}sentation; L{\'e}vy-Khintchin representation; random environment; random probability distribution", } @Article{Bertacchi:2006:ABS, author = "Daniela Bertacchi", title = "Asymptotic Behaviour of the Simple Random Walk on the $2$-dimensional Comb", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "45:1184--45:1203", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-377", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/377", abstract = "We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $ n^{1 / 4}$ for the horizontal projection and $ n^{1 / 2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $ n^{1 / 4}$ and $ n^{1 / 2}$ the horizontal and vertical ones, respectively. The limit process is obtained. With similar techniques the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian Motion; Comb; Generating Function; Maximal Excursion; Random Walk", } @Article{Lifshits:2006:SDG, author = "Mikhail Lifshits and Werner Linde and Zhan Shi", title = "Small Deviations of {Gaussian} Random Fields in {$ L_q $}-Spaces", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "46:1204--46:1233", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-379", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/379", abstract = "We investigate small deviation properties of Gaussian random fields in the space $ L_q(R^N, \mu) $ where $ \mu $ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby ``thin'' measures $ \mu $, i.e., those which are singular with respect to the $N$--dimensional Lebesgue measure; the so-called self-similar measures providing a class of typical examples. For a large class of random fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $ \mu $, called mixed entropy, characterizing size and regularity of $ \mu $. For the particularly interesting case of self-similar measures $ \mu $, the asymptotic behavior of the mixed entropy is evaluated explicitly. As a consequence, we get the asymptotic of the small deviation for $N$-parameter fractional Brownian motions with respect to $ L_q(R^N, \mu)$-norms. While the upper estimates for the small deviation probabilities are proved by purely probabilistic methods, the lower bounds are established by analytic tools concerning Kolmogorov and entropy numbers of Holder operators.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "fractal measures; fractional Brownian motion; Gaussian random fields; Kolmogorov numbers; metric entropy", } @Article{Barbour:2006:DSW, author = "Andrew Barbour and Gesine Reinert", title = "Discrete small world networks", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "47:1234--47:1283", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-381", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/381", abstract = "Small world models are networks consisting of many local links and fewer long range `shortcuts', used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point network distances. We establish approximations to the distribution of the graph distance in a discrete ring network with extra random links, and compare the results to those for simpler models, in which the extra links have zero length and the ring is continuous.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Small-world networks, shortest path length, branching process", } @Article{Su:2006:GFC, author = "Zhonggen Su", title = "{Gaussian} Fluctuations in Complex Sample Covariance Matrices", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "48:1284--48:1320", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-378", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/378", abstract = "Let $ X = (X_{i, j})_{m \times n}, m \ge n $, be a complex Gaussian random matrix with mean zero and variance $ \frac 1 n $, let $ S = X^*X $ be a sample covariance matrix. In this paper we are mainly interested in the limiting behavior of eigenvalues when $ \frac m n \rightarrow \gamma \ge 1 $ as $ n \rightarrow \infty $. Under certain conditions on $k$, we prove the central limit theorem holds true for the $k$-th largest eigenvalues $ \lambda_{(k)}$ as $k$ tends to infinity as $ n \rightarrow \infty $. The proof is largely based on the Costin--Lebowitz--Soshnikov argument and the asymptotic estimates for the expectation and variance of the number of eigenvalues in an interval. The standard technique for the RH problem is used to compute the exact formula and asymptotic properties for the mean density of eigenvalues. As a by-product, we obtain a convergence speed of the mean density of eigenvalues to the Marchenko--Pastur distribution density under the condition $ | \frac m n - \gamma | = O(\frac 1 n)$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Central limit theorem; Eigenvalues; RH problems; Sample covariance matrices; the Costin--Lebowitz--Soshnikov theorem", } @Article{Chaumont:2006:LEP, author = "Loic Chaumont and Juan Carlos Pardo Millan", title = "The Lower Envelope of Positive Self-Similar {Markov} Processes", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "49:1321--49:1341", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-382", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/382", abstract = "We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $ + \infty $. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretzky and Erdos (1951), Motoo (1958), and Rivero (2003).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Self-similar Markov process, L'evy process, Lamperti representation, last passage time, time reversal, integral test, law of the iterated logarithm", } @Article{Johansson:2006:EGM, author = "Kurt Johansson and Eric Nordenstam", title = "Eigenvalues of {GUE} Minors", journal = j-ELECTRON-J-PROBAB, volume = "11", pages = "50:1342--50:1371", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v11-370", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See erratum \cite{Johansson:2007:EEG}.", URL = "http://ejp.ejpecp.org/article/view/370", abstract = "Consider an infinite random matrix $ H = (h_{ij})_{0 < i, j} $ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $ H_i = (h_{rs})_{1 \leq r, s \leq i} $ and let the $j$:th largest eigenvalue of $ H_i$ be $ \mu^i_j$. We show that the configuration of all these eigenvalues $ (i, \mu_j^i)$ form a determinantal point process on $ \mathbb {N} \times \mathbb {R}$.\par Furthermore we show that this process can be obtained as the scaling limit in random tilings of the Aztec diamond close to the boundary. We also discuss the corresponding limit for random lozenge tilings of a hexagon.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random matrices; Tiling problems", } @Article{Bass:2007:FPR, author = "Richard Bass and Jay Rosen", title = "Frequent Points for Random Walks in Two Dimensions", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "1:1--1:46", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-388", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/388", abstract = "For a symmetric random walk in $ Z^2 $ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erd{\H{o}}s--Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time $n$. Among the tools we use are Harnack inequalities and Green's function estimates for random walks with unbounded jumps; some of these are of independent interest.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random walks, Green's functions, Harnack inequalities, frequent points", } @Article{Ivanoff:2007:CCP, author = "B. Gail Ivanoff and Ely Merzbach and Mathieu Plante", title = "A Compensator Characterization of Point Processes on Topological Lattices", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "2:47--2:74", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-390", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/390", abstract = "We resolve the longstanding question of how to define the compensator of a point process on a general partially ordered set in such a way that the compensator exists, is unique, and characterizes the law of the process. We define a family of one-parameter compensators and prove that this family is unique in some sense and characterizes the finite dimensional distributions of a totally ordered point process. This result can then be applied to a general point process since we prove that such a process can be embedded into a totally ordered point process on a larger space. We present some examples, including the partial sum multiparameter process, single line point processes, multiparameter renewal processes, and obtain a new characterization of the two-parameter Poisson process", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "point process, compensator, partial order, single jump process, partial sum process, adapted random set, renewal process, Poisson process, multiparameter martingale", } @Article{Luczak:2007:ADC, author = "Malwina Luczak and Colin McDiarmid", title = "Asymptotic distributions and chaos for the supermarket model", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "3:75--3:99", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-391", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/391", abstract = "In the supermarket model there are $n$ queues, each with a unit rate server. Customers arrive in a Poisson process at rate $ \lambda n$, where $ 0 < \lambda < 1$. Each customer chooses $ d \geq 2$ queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as $ n \to \infty $. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order $ 1 / n$ and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most $ 1 / n$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Supermarket model, join the shortest queue, random choices, power of two choices, load balancing, equilibrium, concentration of measure, law of large numbers, chaos", } @Article{Mendez:2007:ETS, author = "Pedro Mendez", title = "Exit Times of Symmetric Stable Processes from Unbounded Convex Domains", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "4:100--4:121", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-393", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/393", abstract = "We provide several inequalities on the asymptotic behavior of the harmonic measure of the first exit position of a $d$-dimensional symmetric stable process from a unbounded convex domain. Our results on the harmonic measure will determine the asymptotic behavior of the distributions of the first exit time from the domain. These inequalities are given in terms of the growth of the in radius of the cross sections of the domain.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stable process, exit times, unbounded domains", } @Article{Heveling:2007:PSC, author = "Matthias Heveling and Gunter Last", title = "Point shift characterization of {Palm} measures on {Abelian} groups", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "5:122--5:137", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-394", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/394", abstract = "Our first aim in this paper is to characterize Palm measures of stationary point processes through point stationarity. This generalizes earlier results from the Euclidean case to the case of an Abelian group. While a stationary point process looks statistically the same from each site, a point stationary point process looks statistically the same from each of its points. Even in the Euclidean case our proof will simplify some of the earlier arguments. A new technical result of some independent interest is the existence of a complete countable family of matchings. Using a change of measure we will generalize our results to discrete random measures. In the Euclidean case we will finally treat general random measures by means of a suitable approximation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "point process, random measure, stationarity, point-stationarity, Palm measure, matching, bijective point map", } @Article{Uchiyama:2007:AEG, author = "Kouhei Uchiyama", title = "Asymptotic Estimates of the {Green} Functions and Transition Probabilities for {Markov} Additive Processes", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "6:138--6:180", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-396", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/396", abstract = "In this paper we shall derive asymptotic expansions of the Green function and the transition probabilities of Markov additive (MA) processes $ (\xi_n, S_n) $ whose first component satisfies Doeblin's condition and the second one takes valued in $ Z^d $. The derivation is based on a certain perturbation argument that has been used in previous works in the same context. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important for instance in computation of the harmonic measures of a half space for certain models. We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be clarified and it will be shown that in a simple manner the process, if not degenerate, are transformed to another one that satisfies Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if the MA processes is irreducible as a Markov process, then the Green function is expanded quite similarly to that of a classical random walk on $ Z^d $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "asymptotic expansion, harmonic analysis, semi-Markov process, random walk with internal states, perturbation, aperiodicity, ergodic, Doeblin's condition", } @Article{Pipiras:2007:IRP, author = "Vladas Pipiras and Murad Taqqu", title = "Integral representations of periodic and cyclic fractional stable motions", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "7:181--7:206", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-395", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/395", abstract = "Stable non-Gaussian self-similar mixed moving averages can be decomposed into several components. Two of these are the periodic and cyclic fractional stable motions which are the subject of this study. We focus on the structure of their integral representations and show that the periodic fractional stable motions have, in fact, a canonical representation. We study several examples and discuss questions of uniqueness, namely how to determine whether two given integral representations of periodic or cyclic fractional stable motions give rise to the same process.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stable, self-similar processes with stationary increments, mixed moving averages, periodic and cyclic flows, cocycles, semi-additive functionals", } @Article{Coquet:2007:CVO, author = "Fran{\c{c}}ois Coquet and Sandrine Toldo", title = "Convergence of values in optimal stopping and convergence of optimal stopping times", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "8:207--8:228", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-288", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/288", abstract = "Under the hypothesis of convergence in probability of a sequence of c{\`a}dl{\`a}g processes $ (X^n) $ to a c{\`a}dl{\`a}g process $X$, we are interested in the convergence of corresponding values in optimal stopping and also in the convergence of optimal stopping times. We give results under hypothesis of inclusion of filtrations or convergence of filtrations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Convergence of filtrations; Convergence of stochastic processes; Convergence of stopping times.; Optimal stopping times; Values in optimal stopping", } @Article{Labarbe:2007:ABR, author = "Jean-Maxime Labarbe and Jean-Fran{\c{c}}ois Marckert", title = "Asymptotics of {Bernoulli} random walks, bridges, excursions and meanders with a given number of peaks", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "9:229--9:261", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-397", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/397", abstract = "A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being +1 or -1, equally likely. The other families quoted in the title are Bernoulli random walks under various conditions. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the main effect of setting the number of peaks is to change the order of magnitude of the trajectories. The counting process of the peaks, that encodes the repartition of the peaks in the trajectories, is also studied. It is shown that suitably normalized, it converges to a Brownian bridge which is independent of the limiting trajectory. Applications in terms of plane trees and parallelogram polyominoes are provided, as well as an application to the ``comparison'' between runs and Kolmogorov--Smirnov statistics.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Bernoulli random walks; bridge; Brownian meander; excursion; peaks; Weak convergence", } @Article{Ganapathy:2007:RM, author = "Murali Ganapathy", title = "Robust Mixing", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "10:262--10:299", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-398", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/398", abstract = "In this paper, we develop a new ``robust mixing'' framework for reasoning about adversarially modified Markov Chains (AMMC). Let $ \mathbb {P} $ be the transition matrix of an irreducible Markov Chain with stationary distribution $ \pi $. An adversary announces a sequence of stochastic matrices $ \{ \mathbb {A}_t \}_{t > 0} $ satisfying $ \pi \mathbb {A}_t = \pi $. An AMMC process involves an application of $ \mathbb {P} $ followed by $ \mathbb {A}_t $ at time $t$. The robust mixing time of an ergodic Markov Chain $ \mathbb {P}$ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains.\par {\bf Non-Markovian card shuffling processes}: The random-to-cyclic transposition process is a {\em non-Markovian} card shuffling process, which at time $t$, exchanges the card at position $ L_t := t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed a lower bound of $ (0.0345 + o(1))n \log n$ for the mixing time of the random-to-cyclic transposition process. They also considered a generalization of this process where the choice of $ L_t$ is adversarial, and proved an upper bound of $ C n \log n + O(n)$ (with $ C \approx 4 \times 10^5$) on the mixing time. We reduce the constant to $1$ by showing that the random-to-top transposition chain ({\em a Markov Chain}) has robust mixing time $ \leq n \log n + O(n)$ when the adversarial strategies are limited to holomorphic strategies, i.e., those strategies which preserve the symmetry of the underlying Markov Chain. We also show a $ O(n \log^2 n)$ bound on the robust mixing time of the lazy random-to-top transposition chain when the adversary is not limited to holomorphic strategies.\par {\bf Reversible liftings}: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain $ \mathbb {P}$, any reversible lifting $ \mathbb {Q}$ of $ \mathbb {P}$ must satisfy $ \mathcal {T}(\mathbb {P}) \leq \mathcal {T}(\mathbb {Q}) \log (1 / \pi_*)$ where $ \pi_*$ is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that $ \mathcal {T}(\mathbb {Q}) \geq r(\mathbb {P})$ where $ r(\mathbb {P})$ is the relaxation time of $ \mathbb {P}$. This gives an alternate proof of the reversible lifting result and helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov Chains, Robust mixing time, Reversible lifting, random-to-cyclic transposition, non-Markovian processes", } @Article{Lachal:2007:FHT, author = "Aim{\'e} Lachal", title = "First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation {$ \partial u / \partial t = \pm \partial^N u / \partial x^N $}", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "11:300--11:353", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-399", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/399", abstract = "Consider the high-order heat-type equation $ \partial u / \partial t = \pm \partial^N u / \partial x^N $ for an integer $ N > 2 $ and introduce the related Markov pseudo-process $ (X(t))_{t \ge 0} $. In this paper, we study several functionals related to $ (X(t))_{t \ge 0} $: the maximum $ M(t) $ and minimum $ m(t) $ up to time $t$; the hitting times $ \tau_a^+$ and $ \tau_a^-$ of the half lines $ (a, + \infty)$ and $ ( - \infty, a)$ respectively. We provide explicit expressions for the distributions of the vectors $ (X(t), M(t))$ and $ (X(t), m(t))$, as well as those of the vectors $ (\tau_a^+, X(\tau_a^+))$ and $ (\tau_a^-, X(\tau_a^-))$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "first hitting time and place; joint distribution of the process and its maximum/minimum; Multipoles; pseudo-process; Spitzer's identity", } @Article{Valle:2007:EIT, author = "Glauco Valle", title = "Evolution of the interfaces in a two dimensional {Potts} model", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "12:354--12:386", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-346", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/346", abstract = "We investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diffusive scaling the shape of the interfaces converges in probability to the solution of a non-linear parabolic equation. This Law of Large Numbers is obtained from the Hydrodynamic limit of a coupling between an exclusion process and an inhomogeneous one dimensional zero range process with asymmetry at the origin.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Exclusion Processes, Interface Dynamics, Hydrodynamic limit", } @Article{Masiero:2007:RPT, author = "Federica Masiero", title = "Regularizing Properties for Transition Semigroups and Semilinear Parabolic Equations in {Banach} Spaces", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "13:387--13:419", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-401", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/401", abstract = "We study regularizing properties for transition semigroups related to Ornstein Uhlenbeck processes with values in a Banach space $E$ which is continuously and densely embedded in a real and separable Hilbert space $H$. Namely we study conditions under which the transition semigroup maps continuous and bounded functions into differentiable functions. Via a Girsanov type theorem such properties extend to perturbed Ornstein Uhlenbeck processes. We apply the results to solve in mild sense semilinear versions of Kolmogorov equations in $E$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Banach spaces.; Ornstein--Uhlenbeck and perturbed Ornstein--Uhlenbeck transition semigroups; parabolic equations; regularizing properties", } @Article{Lambert:2007:QSD, author = "Amaury Lambert", title = "Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "14:420--14:446", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-402", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/402", abstract = "We consider continuous-state branching (CB) processes which become extinct (i.e., hit 0) with positive probability. We characterize all the quasi-stationary distributions (QSD) for the CB-process as a stochastically monotone family indexed by a real number. We prove that the minimal element of this family is the so-called Yaglom quasi-stationary distribution, that is, the limit of one-dimensional marginals conditioned on being nonzero. Next, we consider the branching process conditioned on not being extinct in the distant future, or $Q$-process, defined by means of Doob $h$-transforms. We show that the $Q$-process is distributed as the initial CB-process with independent immigration, and that under the $ L \log L$ condition, it has a limiting law which is the size-biased Yaglom distribution (of the CB-process). More generally, we prove that for a wide class of nonnegative Markov processes absorbed at 0 with probability 1, the Yaglom distribution is always stochastically dominated by the stationary probability of the $Q$-process, assuming that both exist. Finally, in the diffusion case and in the stable case, the $Q$-process solves a SDE with a drift term that can be seen as the instantaneous immigration.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Continuous-state branching process; h-transform; immigration; L{\'e}vy process; Q-process; quasi-stationary distribution; size-biased distribution; stochastic differential equations; Yaglom theorem", } @Article{Giovanni:2007:SCG, author = "Peccati Giovanni and Murad Taqqu", title = "Stable convergence of generalized {$ L^2 $} stochastic integrals and the principle of conditioning", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "15:447--15:480", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-404", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/404", abstract = "We consider generalized adapted stochastic integrals with respect to independently scattered random measures with second moments, and use a decoupling technique, formulated as a \flqq principle of conditioning\frqq, to study their stable convergence towards mixtures of infinitely divisible distributions. The goal of this paper is to develop the theory. Our results apply, in particular, to Skorohod integrals on abstract Wiener spaces, and to multiple integrals with respect to independently scattered and finite variance random measures. The first application is discussed in some detail in the final section of the present work, and further extended in a companion paper (Peccati and Taqqu (2006b)). Applications to the stable convergence (in particular, central limit theorems) of multiple Wiener--It{\^o} integrals with respect to independently scattered (and not necessarily Gaussian) random measures are developed in Peccati and Taqqu (2006a, 2007). The present work concludes with an example involving quadratic Brownian functionals.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Decoupling; Generalized stochastic integrals; Independently scattered measures; multiple Poisson integrals; Principle of conditioning; Resolutions of the identity; Skorohod integrals; Stable convergence; Weak convergence", } @Article{Galvin:2007:SCR, author = "David Galvin", title = "Sampling $3$-colourings of regular bipartite graphs", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "16:481--16:497", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-403", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/403", abstract = "We show that if $ G = (V, E) $ is a regular bipartite graph for which the expansion of subsets of a single parity of $V$ is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise non-adjacent vertices), and if $M$ is a Markov chain on the set of proper 3-colourings of $G$ which updates the colour of at most $ c|V|$ vertices at each step and whose stationary distribution is uniform, then for $ c < .22$ and $d$ sufficiently large the convergence to stationarity of $M$ is (essentially) exponential in $ |V|$. In particular, if $G$ is the $d$-dimensional hypercube $ Q_d$ (the graph on vertex set $ \{ 0, 1 \}^d$ in which two strings are adjacent if they differ on exactly one coordinate) then the convergence to stationarity of the well-known Glauber (single-site update) dynamics is exponentially slow in $ 2^d / (\sqrt {d} \log d)$. A combinatorial corollary of our main result is that in a uniform 3-colouring of $ Q_d$ there is an exponentially small probability (in $ 2^d$) that there is a colour $i$ such the proportion of vertices of the even subcube coloured $i$ differs from the proportion of the odd subcube coloured $i$ by at most $ .22$. Our proof combines a conductance argument with combinatorial enumeration methods.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Mixing time, 3-colouring, Potts model, conductance, Glauber dynamics, discrete hypercube", } @Article{Evans:2007:ECE, author = "Steven Evans and Tye Lidman", title = "Expectation, Conditional Expectation and Martingales in Local Fields", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "17:498--17:515", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-405", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/405", abstract = "We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $ L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $ L^\infty $ is the appropriate counterpart of $ L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $ L^\infty $ onto the constants.\par Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field. However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $ L^\infty $ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "conditional expectation; expectation; local field; martingale; martingale convergence; optional sampling; projection", } @Article{Gartner:2007:ICS, author = "J{\"u}rgen G{\"a}rtner and Frank den Hollander and Gregory Maillard", title = "Intermittency on catalysts: symmetric exclusion", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "18:516--18:573", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-407", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/407", abstract = "We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a ``reactant'' under the influence of a ``catalyst''. In this paper we focus on the case where the random field is an exclusion process with a symmetric random walk transition kernel, starting from Bernoulli equilibrium. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents when the diffusion constant tends to infinity, which is controlled by moderate deviations of the random field requiring a delicate expansion argument.\par In G{\"a}rtner and den Hollander [10] the case of a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "catalytic random medium; exclusion processes; intermittency; Lyapunov exponents; Parabolic Anderson model", } @Article{Warren:2007:DBM, author = "Jon Warren", title = "{Dyson}'s {Brownian} motions, intertwining and interlacing", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "19:573--19:590", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-406", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/406", abstract = "A reflected Brownian motion in the Gelfand--Tsetlin cone is used to construct Dyson's process of non-colliding Brownian motions. The key step of the construction is to consider two interlaced families of Brownian paths with paths belonging to the second family reflected off paths belonging to the first. Such families of paths are known to arise in the Arratia flow of coalescing Brownian motions. A determinantal formula for the distribution of coalescing Brownian motions is presented.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "coalescing Brownian motions; Gelfand--Tsetlin cone.; intertwining; non-colliding Brownian motions", } @Article{Benjamini:2007:RSR, author = "Itai Benjamini and Roey Izkovsky and Harry Kesten", title = "On the Range of the Simple Random Walk Bridge on Groups", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "20:591--20:612", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-408", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/408", abstract = "Let $G$ be a vertex transitive graph. A study of the range of simple random walk on $G$ and of its bridge is proposed. While it is expected that on a graph of polynomial growth the sizes of the range of the unrestricted random walk and of its bridge are the same in first order, this is not the case on some larger graphs such as regular trees. Of particular interest is the case when $G$ is the Cayley graph of a group. In this case we even study the range of a general symmetric (not necessarily simple) random walk on $G$. We hope that the few examples for which we calculate the first order behavior of the range here will help to discover some relation between the group structure and the behavior of the range. Further problems regarding bridges are presented.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "range of a bridge; range of random walk", } @Article{Toninelli:2007:CLR, author = "Fabio Lucio Toninelli", title = "Correlation Lengths for Random Polymer Models and for Some Renewal Sequences", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "21:613--21:636", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-414", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/414", abstract = "We consider models of directed polymers interacting with a one-dimensional defect line on which random charges are placed. More abstractly, one starts from renewal sequence on $Z$ and gives a random (site-dependent) reward or penalty to the occurrence of a renewal at any given point of $Z$. These models are known to undergo a delocalization-localization transition, and the free energy $F$ vanishes when the critical point is approached from the localized region. We prove that the quenched correlation length $ \xi $, defined as the inverse of the rate of exponential decay of the two-point function, does not diverge faster than $ 1 / F$. We prove also an exponentially decaying upper bound for the disorder-averaged two-point function, with a good control of the sub-exponential prefactor. We discuss how, in the particular case where disorder is absent, this result can be seen as a refinement of the classical renewal theorem, for a specific class of renewal sequences.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Pinning and Wetting Models, Typical and Average Correlation Lengths, Critical Exponents, Renewal Theory, Exponential Convergence Rates", } @Article{Matzinger:2007:DLP, author = "Heinrich Matzinger and Serguei Popov", title = "Detecting a Local Perturbation in a Continuous Scenery", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "22:637--22:660", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-409", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/409", abstract = "A continuous one-dimensional scenery is a double-infinite sequence of points (thought of as locations of {\em bells}) in $R$. Assume that a scenery $X$ is observed along the path of a Brownian motion in the following way: when the Brownian motion encounters a bell different from the last one visited, we hear a ring. The trajectory of the Brownian motion is unknown, whilst the scenery $X$ is known except in some finite interval. We prove that given only the sequence of times of rings, we can a.s. reconstruct the scenery $X$ entirely. For this we take the scenery$X$ to be a local perturbation of a Poisson scenery $ X'$. We present an explicit reconstruction algorithm. This problem is the continuous analog of the ``detection of a defect in a discrete scenery''. Many of the essential techniques used with discrete sceneries do not work with continuous sceneries.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion, Poisson process, localization test, detecting defects in sceneries seen along random walks", } @Article{Dietz:2007:OLS, author = "Zach Dietz and Sunder Sethuraman", title = "Occupation laws for some time-nonhomogeneous {Markov} chains", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "23:661--23:683", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-413", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/413", abstract = "We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time $n$ is $ I + G / n^z$ where $G$ is a ``generator'' matrix, that is $ G(i, j) > 0$ for $ i, j$ distinct, and $ G(i, i) = - \sum_{k \ne i} G(i, k)$, and $ z > 0$ is a strength parameter. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing schemes. These chains, however, exhibit a trichotomy of occupation behaviors depending on parameters.\par We show that the average occupation or empirical distribution vector up to time $n$, when variously $ 0 < z < 1$, $ z > 1$ or $ z = 1$, converges in probability to a unique ``stationary'' vector $ n_G$, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution $ m_G$ with no atoms and full support on a simplex respectively, as $n$ tends to infinity. This last type of limit can be interpreted as a sort of ``spreading'' between the cases $ 0 < z < 1$ and $ z > 1$.\par In particular, when $G$ is appropriately chosen, $ m_G$ is a Dirichlet distribution, reminiscent of results in Polya urns.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "laws of large numbers, nonhomogeneous, Markov, occupation, reinforcement, Dirichlet distribution", } @Article{Ferrari:2007:QSD, author = "Pablo Ferrari and Nevena Maric", title = "Quasi Stationary Distributions and {Fleming--Viot} Processes in Countable Spaces", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "24:684--24:702", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-415", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/415", abstract = "We consider an irreducible pure jump Markov process with rates $ Q = (q(x, y)) $ on $ \Lambda \cup \{ 0 \} $ with $ \Lambda $ countable and $0$ an absorbing state. A {\em quasi stationary distribution \rm} (QSD) is a probability measure $ \nu $ on $ \Lambda $ that satisfies: starting with $ \nu $, the conditional distribution at time $t$, given that at time $t$ the process has not been absorbed, is still $ \nu $. That is, $ \nu (x) = \nu P_t(x) / (\sum_{y \in \Lambda } \nu P_t(y))$, with $ P_t$ the transition probabilities for the process with rates $Q$.\par A {\em Fleming--Viot} (FV) process is a system of $N$ particles moving in $ \Lambda $. Each particle moves independently with rates $Q$ until it hits the absorbing state $0$; but then instantaneously chooses one of the $ N - 1$ particles remaining in $ \Lambda $ and jumps to its position. Between absorptions each particle moves with rates $Q$ independently.\par Under the condition $ \alpha := \sum_{x \in \Lambda } \inf Q(\cdot, x) > \sup Q(\cdot, 0) := C$ we prove existence of QSD for $Q$; uniqueness has been proven by Jacka and Roberts. When $ \alpha > 0$ the FV process is ergodic for each $N$. Under $ \alpha > C$ the mean normalized densities of the FV unique stationary measure converge to the QSD of $Q$, as $ N \to \infty $; in this limit the variances vanish.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Fleming--Viot process; Quasi stationary distributions", } @Article{vanderHofstad:2007:DRG, author = "Remco van der Hofstad and Gerard Hooghiemstra and Dmitri Znamenski", title = "Distances in Random Graphs with Finite Mean and Infinite Variance Degrees", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "25:703--25:766", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-420", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/420", abstract = "In this paper we study typical distances in random graphs with i.i.d. degrees of which the tail of the common distribution function is regularly varying with exponent $ 1 - \tau $. Depending on the value of the parameter $ \tau $ we can distinct three cases: (i) $ \tau > 3 $, where the degrees have finite variance, (ii) $ \tau \in (2, 3) $, where the degrees have infinite variance, but finite mean, and (iii) $ \tau \in (1, 2) $, where the degrees have infinite mean. The distances between two randomly chosen nodes belonging to the same connected component, for $ \tau > 3 $ and $ \tau \in (1, 2), $ have been studied in previous publications, and we survey these results here. When $ \tau \in (2, 3) $, the graph distance centers around $ 2 \log \log {N} / | \log (\tau - 2)| $. We present a full proof of this result, and study the fluctuations around this asymptotic means, by describing the asymptotic distribution. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.\par The random graphs studied here can serve as models for complex networks where degree power laws are observed; this is illustrated by comparing the typical distance in this model to Internet data, where a degree power law with exponent $ \tau \approx 2.2 $ is observed for the so-called Autonomous Systems (AS) graph", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Branching processes, configuration model, coupling, graph distance", } @Article{Gnedin:2007:CR, author = "Alexander Gnedin", title = "The Chain Records", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "26:767--26:786", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-410", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/410", abstract = "Chain records is a new type of multidimensional record. We discuss how often the chain records occur when the background sampling is from the unit cube with uniform distribution (or, more generally, from an arbitrary continuous product distribution in d dimensions). Extensions are given for sampling from more general spaces with a self-similarity property.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "chains; Ewens partition; multidimensional records; random orders", } @Article{Feng:2007:LDD, author = "Shui Feng", title = "Large Deviations for {Dirichlet} Processes and {Poisson--Dirichlet} Distribution with Two Parameters", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "27:787--27:807", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-417", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/417", abstract = "Large deviation principles are established for the two-parameter Poisson--Dirichlet distribution and two-parameter Dirichlet process when parameter $ \theta $ approaches infinity. The motivation for these results is to understand the differences in terms of large deviations between the two-parameter models and their one-parameter counterparts. New insight is obtained about the role of the second parameter $ \alpha $ through a comparison with the corresponding results for the one-parameter Poisson--Dirichlet distribution and Dirichlet process.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Dirichlet processes; GEM representation; large deviations; Poisson--Dirichlet distribution", } @Article{Taylor:2007:CAP, author = "Jesse Taylor", title = "The Common Ancestor Process for a {Wright--Fisher} Diffusion", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "28:808--28:847", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-418", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/418", abstract = "Rates of molecular evolution along phylogenetic trees are influenced by mutation, selection and genetic drift. Provided that the branches of the tree correspond to lineages belonging to genetically isolated populations (e.g., multi-species phylogenies), the interplay between these three processes can be described by analyzing the process of substitutions to the common ancestor of each population. We characterize this process for a class of diffusion models from population genetics theory using the structured coalescent process introduced by Kaplan et al. (1988) and formalized in Barton et al. (2004). For two-allele models, this approach allows both the stationary distribution of the type of the common ancestor and the generator of the common ancestor process to be determined by solving a one-dimensional boundary value problem. In the case of a Wright--Fisher diffusion with genic selection, this solution can be found in closed form, and we show that our results complement those obtained by Fearnhead (2002) using the ancestral selection graph. We also observe that approximations which neglect recurrent mutation can significantly underestimate the exact substitution rates when selection is strong. Furthermore, although we are unable to find closed-form expressions for models with frequency-dependent selection, we can still solve the corresponding boundary value problem numerically and then use this solution to calculate the substitution rates to the common ancestor. We illustrate this approach by studying the effect of dominance on the common ancestor process in a diploid population. Finally, we show that the theory can be formally extended to diffusion models with more than two genetic backgrounds, but that it leads to systems of singular partial differential equations which we have been unable to solve.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Common-ancestor process; diffusion process; genetic drift; selection; structured coalescent; substitution rates", } @Article{Gautier:2007:SNS, author = "Eric Gautier", title = "Stochastic Nonlinear {Schr{\"o}dinger} Equations Driven by a Fractional Noise. {Well}-Posedness, Large Deviations and Support", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "29:848--29:861", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-416", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/416", abstract = "We consider stochastic nonlinear Schrodinger equations driven by an additive noise. The noise is fractional in time with Hurst parameter $ H \in (0, 1) $ and colored in space with a nuclear space correlation operator. We study local well-posedness. Under adequate assumptions on the initial data, the space correlations of the noise and for some saturated nonlinearities, we prove sample path large deviations and support results in a space of Holder continuous in time until blow-up paths. We consider Kerr nonlinearities when $ H > 1 / 2 $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "fractional Brownian motion; Large deviations; nonlinear Schrodinger equation; stochastic partial differential equations", } @Article{Hambly:2007:NVP, author = "Ben Hambly and Liza Jones", title = "Number variance from a probabilistic perspective: infinite systems of independent {Brownian} motions and symmetric alpha stable processes", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "30:862--30:887", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-419", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See erratum \cite{Hambly:2009:ENV}.", URL = "http://ejp.ejpecp.org/article/view/419", abstract = "Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric alpha-stable processes are used to construct explicit new examples of processes which exhibit both divergent and saturating number variance behaviour. We derive a general expression for the number variance for the spatial particle configurations arising from these systems and this enables us to deduce various limiting distribution results for the fluctuations of the associated counting functions. In particular, knowledge of the number variance allows us to introduce and characterize a novel family of centered, long memory Gaussian processes. We obtain fractional Brownian motion as a weak limit of these constructed processes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "controlled variability; fractional Brownian motion; functional limits; Gaussian fluctuations; Gaussian processes; long memory; Number variance; symmetric alpha-stable processes", } @Article{Weill:2007:ARB, author = "Mathilde Weill", title = "Asymptotics for Rooted Bipartite Planar Maps and Scaling Limits of Two-Type Spatial Trees", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "31:862--31:925", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-425", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/425", abstract = "We prove some asymptotic results for the radius and the profile of large random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted bipartite planar maps and certain two-type trees with positive labels, we derive our results from a conditional limit theorem for two-type spatial trees. Finally we apply our estimates to separating vertices of bipartite planar maps: with probability close to one when n tends to infinity, a random $ 2 k$-angulation with n faces has a separating vertex whose removal disconnects the map into two components each with size greater that $ n^{1 / 2 - \varepsilon }$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Conditioned Brownian snake; Planar maps; two-type Galton--Watson trees", } @Article{Benjamini:2007:RGH, author = "Itai Benjamini and Ariel Yadin and Amir Yehudayoff", title = "Random Graph-Homomorphisms and Logarithmic Degree", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "32:926--32:950", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-427", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/427", abstract = "A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph $G$ to the infinite line $Z$. It is shown that if the maximal degree of $G$ is `sub-logarithmic', then the range of such a homomorphism is super-constant.\par Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs $ C_{n, k}$ (which is the tensor product of the $n$-cycle and a complete graph, with self-loops, of size $k$). That is, given any function $ \psi (n)$ tending to infinity, the range of a typical homomorphism of $ C_{n, k}$ is super-constant for $ k = 2 \log (n) - \psi (n)$, and is $3$ for $ k = 2 \log (n) + \psi (n)$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Kurtz:2007:YWE, author = "Thomas Kurtz", title = "The {Yamada--Watanabe--Engelbert} theorem for general stochastic equations and inequalities", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "33:951--33:965", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-431", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/431", abstract = "A general version of the Yamada--Watanabe and Engelbert results relating existence and uniqueness of strong and weak solutions for stochastic equations is given. The results apply to a wide variety of stochastic equations including classical stochastic differential equations, stochastic partial differential equations, and equations involving multiple time transformations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "weak solution, strong solution, pathwise uniqueness, stochastic differential equations, stochastic partial differential equations, multidimensional index", } @Article{Major:2007:MVB, author = "Peter Major", title = "On a Multivariate Version of {Bernstein}'s Inequality", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "34:966--34:988", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-430", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/430", abstract = "We prove such a multivariate version of Bernstein's inequality about the tail distribution of degenerate $U$-statistics which is an improvement of some former results. This estimate will be compared with an analogous bound about the tail distribution of multiple Wiener--It{\^o} integrals. Their comparison shows that our estimate is sharp. The proof is based on good estimates about high moments of degenerate $U$-statistics. They are obtained by means of a diagram formula which enables us to express the product of degenerate $U$-statistics as the sum of such expressions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Bernstein inequality, (degenerate) U-statistics, Wiener--It{\^o} integrals, diagram formula, moment estimates", } @Article{Penrose:2007:GLR, author = "Mathew Penrose", title = "{Gaussian} Limts for Random Geometric Measures", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "35:989--35:1035", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-429", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/429", abstract = "Given $n$ independent random marked $d$-vectors $ X_i$ with a common density, define the measure $ \nu_n = \sum_i \xi_i $, where $ \xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $ X_i$. Technically, this means here that $ \xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $ R^d$, we give a central limit theorem for $ \nu_n(f)$, and deduce weak convergence of $ \nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and $k$-nearest neighbours graph.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random measures", } @Article{Turova:2007:CPT, author = "Tatyana Turova", title = "Continuity of the percolation threshold in randomly grown graphs", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "36:1036--36:1047", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-436", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/436", abstract = "We consider various models of randomly grown graphs. In these models the vertices and the edges accumulate within time according to certain rules. We study a phase transition in these models along a parameter which refers to the mean life-time of an edge. Although deleting old edges in the uniformly grown graph changes abruptly the properties of the model, we show that some of the macro-characteristics of the graph vary continuously. In particular, our results yield a lower bound for the size of the largest connected component of the uniformly grown graph.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "branching processes; Dynamic random graphs; phase transition", } @Article{Johansson:2007:EEG, author = "Kurt Johansson and Eric Nordenstam", title = "Erratum to {``Eigenvalues of GUE Minors''}", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "37:1048--37:1051", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-816", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See \cite{Johansson:2006:EGM}.", URL = "http://ejp.ejpecp.org/article/view/816", abstract = "In the paper \url{http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1647}, two expressions for the so called GUE minor kernel are presented, one in definition 1.2 and one in the formulas (5.6) and (5.7). The expressions given in (5.6) and (5.7) are correct, but the expression in definition 1.2 of the paper has to be modified in the case $ r > s $. The proof of the equality of the two expressions for the GUE minor kernel given in the paper was based on lemma 5.6 which is not correct since some terms in the expansion are missing. The correct expansion is given in lemma 1.2 below.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Arias-Castro:2007:IRH, author = "Ery Arias-Castro", title = "Interpolation of Random Hyperplanes", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "38:1052--38:1071", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-435", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/435", abstract = "Let $ \{ (Z_i, W_i) \colon i = 1, \dots, n \} $ be uniformly distributed in $ [0, 1]^d \times \mathbb {G}(k, d) $, where $ \mathbb {G}(k, d) $ denotes the space of $k$-dimensional linear subspaces of $ \mathbb {R}^d$. For a differentiable function $ f \colon [0, 1]^k \rightarrow [0, 1]^d$, we say that $f$ interpolates $ (z, w) \in [0, 1]^d \times \mathbb {G}(k, d)$ if there exists $ x \in [0, 1]^k$ such that $ f(x) = z$ and $ \vec {f}(x) = w$, where $ \vec {f}(x)$ denotes the tangent space at $x$ defined by $f$. For a smoothness class $ {\cal F}$ of Holder type, we obtain probability bounds on the maximum number of points a function $ f \in {\cal F}$ interpolates.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Grassmann Manifold; Haar Measure; Kolmogorov Entropy; Pattern Recognition", } @Article{Bobkov:2007:LDI, author = "Sergey Bobkov", title = "Large deviations and isoperimetry over convex probability measures with heavy tails", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "39:1072--39:1100", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-440", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/440", abstract = "Large deviations and isoperimetric inequalities are considered for probability distributions, satisfying convexity conditions of the Brunn--Minkowski-type", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Large deviations, convex measures, dilation of sets, transportation of mass, Khinchin-type, isoperimetric, weak Poincar{\'e}, Sobolev-type inequalities", } @Article{Griffiths:2007:RIA, author = "Robert Griffiths and Dario Spano", title = "Record Indices and Age-Ordered Frequencies in Exchangeable {Gibbs} Partitions", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "40:1101--40:1130", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-434", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/434", abstract = "The frequencies of an exchangeable Gibbs random partition of the integers (Gnedin and Pitman 2005) are considered in their age-order, i.e., their size-biased order. We study their dependence on the sequence of record indices (i.e., the least elements) of the blocks of the partition. In particular we show that, conditionally on the record indices, the distribution of the age-ordered frequencies has a left-neutral stick-breaking structure. Such a property in fact characterizes the Gibbs family among all exchangeable partitions and leads to further interesting results on: (i) the conditional Mellin transform of the $k$-th oldest frequency given the $k$-th record index, and (ii) the conditional distribution of the first $k$ normalized frequencies, given their sum and the $k$-th record index; the latter turns out to be a mixture of Dirichlet distributions. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens' partitions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Exchangeable Gibbs Partitions, GEM distribution, Age-ordered frequencies, Beta-Stacy distribution, Neutral distributions, Record indices", } @Article{Maida:2007:LDL, author = "Myl{\`e}ne Maida", title = "Large deviations for the largest eigenvalue of rank one deformations of {Gaussian} ensembles", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "41:1131--41:1150", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-438", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/438", abstract = "We establish a large deviation principle for the largest eigenvalue of a rank one deformation of a matrix from the GUE or GOE. As a corollary, we get another proof of the phenomenon, well-known in learning theory and finance, that the largest eigenvalue separates from the bulk when the perturbation is large enough. A large part of the paper is devoted to an auxiliary result on the continuity of spherical integrals in the case when one of the matrix is of rank one, as studied in one of our previous works.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "large deviations; random matrices", } @Article{Evans:2007:AEA, author = "Steven Evans and Tye Lidman", title = "Asymptotic Evolution of Acyclic Random Mappings", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "42:1051--42:1180", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-437", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/437", abstract = "An acyclic mapping from an $n$ element set into itself is a mapping $ \varphi $ such that if $ \varphi^k(x) = x$ for some $k$ and $x$, then $ \varphi (x) = x$. Equivalently, $ \varphi^\ell = \varphi^{\ell + 1} = \ldots $ for $ \ell $ sufficiently large. We investigate the behavior as $ n \rightarrow \infty $ of a sequence of a Markov chain on the collection of such mappings. At each step of the chain, a point in the $n$ element set is chosen uniformly at random and the current mapping is modified by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Informal calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and ``space'' rescaling, converge as $ n \rightarrow \infty $ to a real tree ($R$-tree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reflected Brownian bridge. Although we don't prove such a limit theorem, we use Dirichlet form methods to construct a Markov process that is Hunt with respect to a suitable Gromov--Hausdorff-like metric and evolves according to the dynamics suggested by the heuristic arguments. This process is similar to one that appears in earlier work by Evans and Winter as a similarly informal limit of a Markov chain related to the subtree prune and regraft tree (SPR) rearrangements from phylogenetics.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian bridge; Brownian excursion; continuum random tree; Dirichlet form; excursion theory; Gromov--Hausdorff metric; path decomposition; random mapping", } @Article{Darses:2007:TRD, author = "Sebastien Darses and Bruno Saussereau", title = "Time Reversal for Drifted Fractional {Brownian} Motion with {Hurst} Index {$ H > 1 / 2 $}", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "43:1181--43:1211", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-439", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/439", abstract = "Let $X$ be a drifted fractional Brownian motion with Hurst index $ H > 1 / 2$. We prove that there exists a fractional backward representation of $X$, i.e., the time reversed process is a drifted fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when $ H = 1 / 2$. We then apply our result to stochastic differential equations driven by a fractional Brownian motion.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Fractional Brownian motion; Malliavin Calculus.; Time reversal", } @Article{Barthe:2007:IBE, author = "Franck Barthe and Patrick Cattiaux and Cyril Roberto", title = "Isoperimetry between exponential and {Gaussian}", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "44:1212--44:1237", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-441", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/441", abstract = "We study the isoperimetric problem for product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Isoperimetry; Super-Poincar{\'e} inequality", } @Article{Rider:2007:CDP, author = "Brian Rider and Balint Virag", title = "Complex Determinantal Processes and {$ H1 $} Noise", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "45:1238--45:1257", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-446", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/446", abstract = "For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes $ \mathcal Z_\rho $ with intensity $ \rho d \nu $, where $ \nu $ is the corresponding invariant measure. We show that as $ \rho \to \infty $, after centering, these processes converge to invariant $ H^1 $ noise. More precisely, for all functions $ f \in H^1 (\nu) \cap L^1 (\nu) $ the distribution of $ \sum_{z \in \mathcal Z} f(z) - \frac {\rho }{\pi } \int f d \nu $ converges to Gaussian with mean zero and variance $ \frac {1}{4 \pi } \| f \|_{H^1}^2 $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "determinantal process; invariant point process; noise limit; random matrices", } @Article{Neunhauserer:2007:RWI, author = "J{\"o}rg Neunh{\"a}userer", title = "Random walks on infinite self-similar graphs", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "46:1258--46:1275", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-448", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/448", abstract = "We introduce a class of rooted infinite self-similar graphs containing the well known Fibonacci graph and graphs associated with Pisot numbers. We consider directed random walks on these graphs and study their entropy and their limit measures. We prove that every infinite self-similar graph has a random walk of full entropy and that the limit measures of this random walks are absolutely continuous.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "graph; random walk", } @Article{Klass:2007:UAQ, author = "Michael Klass and Krzysztof Nowicki", title = "Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "47:1276--47:1298", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-452", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/452", abstract = "Let $ X_1, X_2, \dots $ be independent and symmetric random variables such that $ S_n = X_1 + \cdots + X_n $ converges to a finite valued random variable $S$ a.s. and let $ S^* = \sup_{1 \leq n \leq \infty } S_n$ (which is finite a.s.). We construct upper and lower bounds for $ s_y$ and $ s_y^*$, the upper $ 1 / y$-th quantile of $ S_y$ and $ S^*$, respectively. Our approximations rely on an explicitly computable quantity $ \underline q_y$ for which we prove that\par $$ \frac 1 2 \underline q_{y / 2} < s_y^* < 2 \underline q_{2y} \quad \text { and } \quad \frac 1 2 \underline q_{ (y / 4) (1 + \sqrt { 1 - 8 / y})} < s_y < 2 \underline q_{2y}. $$ The RHS's hold for $ y \geq 2$ and the LHS's for $ y \geq 94$ and $ y \geq 97$, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Sum of independent rv's, tail distributions, tail distributions, tail probabilities, quantile approximation, Hoffmann--J{\o}rgensen/Klass--Nowicki Inequality", } @Article{Grigorescu:2007:EPM, author = "Ilie Grigorescu and Min Kang", title = "Ergodic Properties of Multidimensional {Brownian} Motion with Rebirth", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "48:1299--48:1322", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-450", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/450", abstract = "In a bounded open region of the $d$ dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin's condition. Both methods admit generalizations to a wide class of processes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Dirichlet Laplacian, Green function, analytic semigroup, jump diffusion", } @Article{Biskup:2007:FCR, author = "Marek Biskup and Timothy Prescott", title = "Functional {CLT} for Random Walk Among Bounded Random Conductances", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "49:1323--49:1348", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-456", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/456", abstract = "We consider the nearest-neighbor simple random walk on $ Z^d $, $ d \ge 2 $, driven by a field of i.i.d. random nearest-neighbor conductances $ \omega_{xy} \in [0, 1] $. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $ \omega $'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $ d \ge 5 $ due to anomalously slow decay of the probability that the walk returns to the starting point at a given time.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random conductance model, invariance principle, corrector, homogenization, heat kernel, percolation, isoperimetry", } @Article{Mytnik:2007:LES, author = "Leonid Mytnik and Jie Xiong", title = "Local extinction for superprocesses in random environments", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "50:1349--50:1378", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-457", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/457", abstract = "We consider a superprocess in a random environment represented by a random measure which is white in time and colored in space with correlation kernel $ g(x, y) $. Suppose that $ g(x, y) $ decays at a rate of $ |x - y|^{- \alpha } $, $ 0 \leq \alpha \leq 2 $, as $ |x - y| \to \infty $. We show that the process, starting from Lebesgue measure, suffers long-term local extinction. If $ \alpha < 2 $, then it even suffers finite time local extinction. This property is in contrast with the classical super-Brownian motion which has a non-trivial limit when the spatial dimension is higher than 2. We also show in this paper that in dimensions $ d = 1, 2 $ superprocess in random environment suffers local extinction for any bounded function $g$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Tykesson:2007:NUC, author = "Johan Tykesson", title = "The number of unbounded components in the {Poisson} {Boolean} model of continuum percolation in hyperbolic space", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "51:1379--51:1401", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-460", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/460", abstract = "We consider the Poisson Boolean model of continuum percolation with balls of fixed radius $R$ in $n$-dimensional hyperbolic space $ H^n$. Let $ \lambda $ be the intensity of the underlying Poisson process, and let $ N_C$ denote the number of unbounded components in the covered region. For the model in any dimension we show that there are intensities such that $ N_C = \infty $ a.s. if $R$ is big enough. In $ H^2$ we show a stronger result: for any $R$ there are two intensities $ \lambda_c$ and $ \lambda_u$ where $ 0 < \lambda_c < \lambda_u < \infty $, such that$ N_C = 0$ for $ \lambda \in [0, \lambda_c]$, $ N_C = \infty $ for $ \lambda \in (\lambda_c, \lambda_u)$ and $ N_C = 1$ for $ \lambda \in [\lambda_u, \infty)$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "continuum percolation; hyperbolic space; phase transitions", } @Article{Hu:2007:EES, author = "Zhishui Hu and John Robinson and Qiying Wang", title = "{Edgeworth} Expansions for a Sample Sum from a Finite Set of Independent Random Variables", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "52:1402--52:1417", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-447", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/447", abstract = "Let $ \{ X_1, \cdots, X_N \} $ be a set of $N$ independent random variables, and let $ S_n$ be a sum of $n$ random variables chosen without replacement from the set $ \{ X_1, \cdots, X_N \} $ with equal probabilities. In this paper we give a one-term Edgeworth expansion of the remainder term for the normal approximation of $ S_n$ under mild conditions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Edgeworth expansion, finite population, sampling without replacement", } @Article{Ankirchner:2007:CVD, author = "Stefan Ankirchner and Peter Imkeller and Goncalo {Dos Reis}", title = "Classical and Variational Differentiability of {BSDEs} with Quadratic Growth", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "53:1418--53:1453", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-462", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/462", abstract = "We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. These results can be applied to systems of forward--backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational sense (Malliavin differentiable).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "BSDE, forward--backward SDE, quadratic growth, differentiability, stochastic calculus of variations, Malliavin calculus, Feynman--Kac formula, BMO martingale, reverse Holder inequality", } @Article{Aldous:2007:PUR, author = "David Aldous and Russell Lyons", title = "Processes on Unimodular Random Networks", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "54:1454--54:1508", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-463", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See errata \cite{Aldous:2017:EPU}.", URL = "http://ejp.ejpecp.org/article/view/463", abstract = "We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Amenability, equivalence relations, infinite graphs, percolation, quasi-transitive, random walks, transitivity, weak convergence, reversibility, trace, stochastic comparison, spanning forests, sofic groups", } @Article{White:2007:PID, author = "David White", title = "Processes with inert drift", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "55:1509--55:1546", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-465", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/465", abstract = "We construct a stochastic process whose drift is a function of the process's local time at a reflecting barrier. The process arose as a model of the interactions of a Brownian particle and an inert particle in a paper by Knight [7]. We construct and give asymptotic results for two different arrangements of inert particles and Brownian particles, and construct the analogous process in higher dimensions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion; local time; Skorohod lemma", } @Article{Gnedin:2007:NCL, author = "Alexander Gnedin and Yuri Yakubovich", title = "On the Number of Collisions in Lambda-Coalescents", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "56:1547--56:1567", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-464", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/464", abstract = "We examine the total number of collisions $ C_n $ in the $ \Lambda $-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $ C_n$ are shown under the assumption of a power-like behaviour of the measure $ \Lambda $ near $0$ with exponent $ 0 < \alpha < 1$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "collisions; Lambda-coalescent; stable limit", } @Article{Feng:2007:GIF, author = "Chunrong Feng and Huaizhong Zhao", title = "A Generalized {It{\^o}}'s Formula in Two-Dimensions and Stochastic {Lebesgue--Stieltjes} Integrals", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "57:1568--57:1599", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-468", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/468", abstract = "In this paper, a generalized It$ {\hat {\rm o}} $'s formula for continuous functions of two-dimensional continuous semimartingales is proved. The formula uses the local time of each coordinate process of the semimartingale, the left space first derivatives and the second derivative $ \nabla_1^- \nabla_2^-f $, and the stochastic Lebesgue--Stieltjes integrals of two parameters. The second derivative $ \nabla_1^- \nabla_2^-f $ is only assumed to be of locally bounded variation in certain variables. Integration by parts formulae are asserted for the integrals of local times. The two-parameter integral is defined as a natural generalization of both the It{\^o} integral and the Lebesgue--Stieltjes integral through a type of It$ {\hat {\rm o }} $ isometry formula.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "continuous semimartingale; generalized It{\^o}'s formula; local time; stochastic Lebesgue--Stieltjes integral", } @Article{Janson:2007:TEB, author = "Svante Janson and Guy Louchard", title = "Tail estimates for the {Brownian} excursion area and other {Brownian} areas", journal = j-ELECTRON-J-PROBAB, volume = "12", pages = "58:1600--58:1632", year = "2007", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v12-471", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/471", abstract = "Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian areas, asymptotics for density functions right tail, double Laplace transform, two-dimensional saddle point method", } @Article{Chaumont:2008:CLP, author = "Lo{\"\i}c Chaumont and Ronald Doney", title = "Corrections to {``On L{\'e}vy processes conditioned to stay positive''}", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "1:1--1:4", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-466", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See \cite{Chaumont:2005:LPC}.", URL = "http://ejp.ejpecp.org/article/view/466", abstract = "We correct two errors of omission in our paper, On L{\'e}vy processes conditioned to stay positive. \url{http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1517&layout=abstract} Electron. J. Probab. {\bf 10}, (2005), no. 28, 948--961. Math. Review 2006h:60079.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "L{\'e}vy process, conditioned to stay positive, weak convergence, excursion measure", } @Article{Kurkova:2008:LES, author = "Irina Kurkova", title = "Local Energy Statistics in Directed Polymers", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "2:5--2:25", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-475", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/475", abstract = "Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as well for directed polymers in random environment. We also show that, under certain conditions, this conjecture holds for directed polymers even if energy levels that grow moderately with the volume of the system are considered.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Directed polymers", } @Article{Chen:2008:CPE, author = "Guan-Yu Chen and Laurent Saloff-Coste", title = "The Cutoff Phenomenon for Ergodic {Markov} Processes", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "3:26--3:78", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-474", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/474", abstract = "We consider the cutoff phenomenon in the context of families of ergodic Markov transition functions. This includes classical examples such as families of ergodic finite Markov chains and Brownian motion on families of compact Riemannian manifolds. We give criteria for the existence of a cutoff when convergence is measured in $ L^p$-norm, $ 1 < p < \infty $. This allows us to prove the existence of a cutoff in cases where the cutoff time is not explicitly known. In the reversible case, for $ 1 < p \leq \infty $, we show that a necessary and sufficient condition for the existence of a max-$ L^p$ cutoff is that the product of the spectral gap by the max-$ L^p$ mixing time tends to infinity. This type of condition was suggested by Yuval Peres. Illustrative examples are discussed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "cutoff phenomenon, ergodic Markov semigroups", } @Article{Miermont:2008:RPR, author = "Gr{\'e}gory Miermont and Mathilde Weill", title = "Radius and profile of random planar maps with faces of arbitrary degrees", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "4:79--4:106", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-478", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/478", abstract = "We prove some asymptotic results for the radius and the profile of large random planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco \& Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton--Watson trees.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian snake; invariance principle; multitype spatial Galton--Watson tree; Random planar map", } @Article{Houdre:2008:CSM, author = "Christian Houdr{\'e} and Hua Xu", title = "Concentration of the Spectral Measure for Large Random Matrices with Stable Entries", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "5:107--5:134", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-482", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/482", abstract = "We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries, in particular, stable or heavy tails ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Spectral Measure, Random Matrices, Infinitely divisibility, Stable Vector, Concentration", } @Article{Fournier:2008:SLS, author = "Nicolas Fournier", title = "Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "6:135--6:156", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-480", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/480", abstract = "We consider a one-dimensional jumping Markov process, solving a Poisson-driven stochastic differential equation. We prove that the law of this process admits a smooth density for all positive times, under some regularity and non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge, our result is the first one including the important case of a non-constant rate of jump. The main difficulty is that in such a case, the process is not smooth as a function of its initial condition. This seems to make impossible the use of Malliavin calculus techniques. To overcome this problem, we introduce a new method, in which the propagation of the smoothness of the density is obtained by analytic arguments.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic differential equations, Jump processes, Regularity of the density", } @Article{Savov:2008:CCR, author = "Mladen Savov", title = "Curve Crossing for the Reflected {L{\'e}vy} Process at Zero and Infinity", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "7:157--7:172", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-483", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/483", abstract = "Let $ R_t = \sup_{0 \leq s \leq t}X_s - X_t $ be a Levy process reflected in its maximum. We give necessary and sufficient conditions for finiteness of passage times above power law boundaries at infinity. Information as to when the expected passage time for $ R_t $ is finite, is given. We also discuss the almost sure finiteness of $ \limsup_{t \to 0}R_t / t^{\kappa } $, for each $ \kappa \geq 0 $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Reflected process, passage times, power law boundaries", } @Article{Baurdoux:2008:MSG, author = "Erik Baurdoux and Andreas Kyprianou", title = "The {McKean} stochastic game driven by a spectrally negative {L{\'e}vy} process", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "8:173--8:197", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-484", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/484", abstract = "We consider the stochastic-game-analogue of McKean's optimal stopping problem when the underlying source of randomness is a spectrally negative L{\'e}vy process. Compared to the solution for linear Brownian motion given in Kyprianou (2004) one finds two new phenomena. Firstly the breakdown of smooth fit and secondly the stopping domain for one of the players `thickens' from a singleton to an interval, at least in the case that there is no Gaussian component.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic games, optimal stopping, pasting principles, fluctuation theory, L'evy processes", } @Article{Fill:2008:TPK, author = "James Fill and David Wilson", title = "Two-Player Knock 'em Down", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "9:198--9:212", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-485", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/485", abstract = "We analyze the two-player game of Knock 'em Down, asymptotically as the number of tokens to be knocked down becomes large. Optimal play requires mixed strategies with deviations of order $ \sqrt {n} $ from the na{\"\i}ve law-of-large numbers allocation. Upon rescaling by $ \sqrt {n} $ and sending $ n \to \infty $, we show that optimal play's random deviations always have bounded support and have marginal distributions that are absolutely continuous with respect to Lebesgue measure.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "game theory; Knock 'em Down; Nash equilibrium", } @Article{Caputo:2008:AEP, author = "Pietro Caputo and Fabio Martinelli and Fabio Toninelli", title = "On the Approach to Equilibrium for a Polymer with Adsorption and Repulsion", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "10:213--10:258", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-486", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/486", abstract = "We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after $L$ steps, $ L \in 2 \mathbb {N}$. In the {\em pinning model} each path $ \eta $ has a weight $ \lambda^{N(\eta)}$, where $ \lambda > 0$ and $ N(\eta)$ is the number of zeros in $ \eta $. When the paths are constrained to be non-negative, the polymer is said to satisfy a hard-wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength $ \lambda $ is varied. In this paper we study a natural ``spin flip'' dynamics for associated to these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (i.e., $ \lambda = 1$ without the wall), where the gap and the mixing time are known to scale as $ L^{-2}$ and $ L^2 \log L$, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for $ \lambda \geq 1$ relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase ($ \lambda < 1$) the gap is shown to be $ O(L^{-5 / 2})$, up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Coupling; Dynamical phase transition; Mixing time; Pinning model; Spectral gap", } @Article{Davydov:2008:SSD, author = "Youri Davydov and Ilya Molchanov and Sergei Zuyev", title = "Strictly stable distributions on convex cones", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "11:259--11:321", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-487", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/487", abstract = "Using the LePage representation, a symmetric alpha-stable random element in Banach space B with alpha from (0, 2) can be represented as a sum of points of a Poisson process in B. This point process is union-stable, i.e., the union of its two independent copies coincides in distribution with the rescaled original point process. This shows that the classical definition of stable random elements is closely related to the union-stability property of point processes. These concepts make sense in any convex cone, i.e., in a semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. We prove that random samples (or binomial point processes) in rather general cones converge in distribution in the vague topology to the union-stable Poisson point process. This convergence holds also in a stronger topology, which implies that the sums of points converge in distribution to the sum of points of the union-stable point process. Since the latter corresponds to a stable law, this yields a limit theorem for normalised sums of random elements with alpha-stable limit for alpha from (0, 1). By using the technique of harmonic analysis on semigroups we characterise distributions of alpha-stable random elements and show how possible values of the characteristic exponent alpha relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. It is shown that several conditions imply that a stable random element admits the LePage representation. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "character; convex cone; Laplace transform; LePage series; L{\'e}vy measure; point process; Poisson process; random measure; random set; semigroup; stable distribution; union-stability", } @Article{Merlet:2008:CTS, author = "Glenn Merlet", title = "Cycle time of stochastic max-plus linear systems", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "12:322--12:340", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-488", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/488", abstract = "We analyze the asymptotic behavior of sequences of random variables defined by an initial condition, a stationary and ergodic sequence of random matrices, and an induction formula involving multiplication is the so-called max-plus algebra. This type of recursive sequences are frequently used in applied probability as they model many systems as some queueing networks, train and computer networks, and production systems. We give a necessary condition for the recursive sequences to satisfy a strong law of large numbers, which proves to be sufficient when the matrices are i.i.d. Moreover, we construct a new example, in which the sequence of matrices is strongly mixing, that condition is satisfied, but the recursive sequence do not converges almost surely.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "law of large numbers; Markov chains; max-plus; products of random matrices; stochastic recursive sequences; subadditivity", } @Article{Lamberton:2008:PBA, author = "Damien Lamberton and Gilles Pag{\`e}s", title = "A penalized bandit algorithm", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "13:341--13:373", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-489", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/489", abstract = "We study a two armed-bandit recursive algorithm with penalty. We show that the algorithm converges towards its ``target'' although it always has a noiseless ``trap''. Then, we elucidate the rate of convergence. For some choices of the parameters, we obtain a central limit theorem in which the limit distribution is characterized as the unique stationary distribution of a Markov process with jumps.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "convergence rate; learning; penalization; stochastic approximation; Two-armed bandit algorithm", } @Article{Berestycki:2008:LBD, author = "Nathanael Berestycki and Rick Durrett", title = "Limiting behavior for the distance of a random walk", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "14:374--14:395", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-490", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/490", abstract = "In this paper we study some aspects of the behavior of random walks on large but finite graphs before they have reached their equilibrium distribution. This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $ n / 2 $. Here, we study the examples of random 3-regular graphs, random adjacent transpositions, and riffle shuffles. In the case of a random 3-regular graph, there is a phase transition where the speed changes from 1/3 to 0 at time $ 3 l o g_2 n $. A similar result is proved for riffle shuffles, where the speed changes from 1 to 0 at time $ l o g_2 n $. Both these changes occur when a distance equal to the average diameter of the graph is reached. However in the case of random adjacent transpositions, the behavior is more complex. We find that there is no phase transition, even though the distance has different scalings in three different regimes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random walk, phase transition, adjacent transpositions, random regular graphs, riffle shuffles", } @Article{Lember:2008:IRR, author = "Jyri Lember and Heinrich Matzinger", title = "Information recovery from randomly mixed-up message text", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "15:396--15:466", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-491", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/491", abstract = "This paper is concerned with finding a fingerprint of a sequence. As input data one uses the sequence which has been randomly mixed up by observing it along a random walk path. A sequence containing order exp (n) bits receives a fingerprint with roughly n bits information. The fingerprint is characteristic for the original sequence. With high probability the fingerprint depends only on the initial sequence, but not on the random walk path.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random walk in random environment; Scenery reconstruction", } @Article{Beghin:2008:PPG, author = "Luisa Beghin", title = "Pseudo-Processes Governed by Higher-Order Fractional Differential Equations", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "16:467--16:485", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-496", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/496", abstract = "We study here a heat-type differential equation of order $n$ greater than two, in the case where the time-derivative is supposed to be fractional. The corresponding solution can be described as the transition function of a pseudoprocess $ \Psi_n$ (coinciding with the one governed by the standard, non-fractional, equation) with a time argument $ \mathcal {T}_{\alpha }$ which is itself random. The distribution of $ \mathcal {T}_{\alpha }$ is presented together with some features of the solution (such as analytic expressions for its moments.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Fractional derivatives; Higher-order heat-type equations; Stable laws.; Wright functions", } @Article{Basdevant:2008:AAF, author = "Anne-Laure Basdevant and Christina Goldschmidt", title = "Asymptotics of the Allele Frequency Spectrum Associated with the {Bolthausen--Sznitman} Coalescent", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "17:486--17:512", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-494", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/494", abstract = "We consider a coalescent process as a model for the genealogy of a sample from a population. The population is subject to neutral mutation at constant rate $ \rho $ per individual and every mutation gives rise to a completely new type. The allelic partition is obtained by tracing back to the most recent mutation for each individual and grouping together individuals whose most recent mutations are the same. The allele frequency spectrum is the sequence $ (N_1 (n), N_2 (n), \ldots, N_n(n)) $, where $ N_k(n) $ is number of blocks of size $k$ in the allelic partition with sample size $n$. In this paper, we prove law of large numbers-type results for the allele frequency spectrum when the coalescent process is taken to be the Bolthausen--Sznitman coalescent. In particular, we show that $ n^{-1}(\log n) N_1 (n) {\stackrel {p}{\rightarrow }} \rho $ and, for $ k \geq 2$, $ n^{-1}(\log n)^2 N_k(n) {\stackrel {p}{\rightarrow }} \rho / (k(k - 1))$ as $ n \to \infty $. Our method of proof involves tracking the formation of the allelic partition using a certain Markov process, for which we prove a fluid limit.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Giacomin:2008:RCR, author = "Giambattista Giacomin", title = "Renewal convergence rates and correlation decay for homogeneous pinning models", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "18:513--18:529", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-497", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/497", abstract = "A class of discrete renewal processes with exponentially decaying inter-arrival distributions coincides with the infinite volume limit of general homogeneous pinning models in their localized phase. Pinning models are statistical mechanics systems to which a lot of attention has been devoted both for their relevance for applications and because they are solvable models exhibiting a non-trivial phase transition. The spatial decay of correlations in these systems is directly mapped to the speed of convergence to equilibrium for the associated renewal processes. We show that close to criticality, under general assumptions, the correlation decay rate, or the renewal convergence rate, coincides with the inter-arrival decay rate. We also show that, in general, this is false away from criticality. Under a stronger assumption on the inter-arrival distribution we establish a local limit theorem, capturing thus the sharp asymptotic behavior of correlations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Criticality; Decay of Correlations; Exponential Tails; Pinning Models; Renewal Theory; Speed of Convergence to Equilibrium", } @Article{Merkl:2008:BRE, author = "Franz Merkl and Silke Rolles", title = "Bounding a Random Environment Bounding a Random Environment for Two-dimensional Edge-reinforced Random Walk", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "19:530--19:565", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-495", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/495", abstract = "We consider edge-reinforced random walk on the infinite two-dimensional lattice. The process has the same distribution as a random walk in a certain strongly dependent random environment, which can be described by random weights on the edges. In this paper, we show some decay properties of these random weights. Using these estimates, we derive bounds for some hitting probabilities of the edge-reinforced random walk.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random environment; Reinforced random walk", } @Article{Daly:2008:UBS, author = "Fraser Daly", title = "Upper Bounds for {Stein}-Type Operators", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "20:566--20:587", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-479", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/479", abstract = "We present sharp bounds on the supremum norm of $ \mathcal {D}^j S h $ for $ j \geq 2 $, where $ \mathcal {D} $ is the differential operator and $S$ the Stein operator for the standard normal distribution. The same method is used to give analogous bounds for the exponential, Poisson and geometric distributions, with $ \mathcal {D}$ replaced by the forward difference operator in the discrete case. We also discuss applications of these bounds to the central limit theorem, simple random sampling, Poisson--Charlier approximation and geometric approximation using stochastic orderings.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "central limit theorem; Poisson--Charlier approximation; Stein's method; Stein-type operator; stochastic ordering", } @Article{Bose:2008:ALM, author = "Arup Bose and Arnab Sen", title = "Another look at the moment method for large dimensional random matrices", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "21:588--21:628", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-501", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/501", abstract = "The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample variance covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) establish the LSD for the random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalent classes and relating the limits of the counts to certain volume calculations.\par We build on their work and present a unified approach. This helps provide relatively short and easy proofs for the LSD of common matrices while at the same time providing insight into the nature of different LSD and their interrelations. By extending these methods we are also able to deal with matrices with appropriate dependent entries.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Bounded Lipschitz metric, large dimensional random matrices, eigenvalues, Wigner matrix, sample variance covariance matrix, Toeplitz matrix, Hankel matrix, circulant matrix, symmetric circulant matrix, reverse circulant matrix, palindromic matrix, limit", } @Article{Conus:2008:NLS, author = "Daniel Conus and Robert Dalang", title = "The Non-Linear Stochastic Wave Equation in High Dimensions", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "22:629--22:670", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-500", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/500", abstract = "We propose an extension of Walsh's classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a driving noise that is white in time and correlated in space. In the particular case of an affine multiplicative noise, we obtain estimates on $p$-th moments of the solution ($ p \geq 1$), and we show that the solution is H{\"o}lder continuous. The H{\"o}lder exponent that we obtain is optimal.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "H{\"o}lder continuity; Martingale measures; moment formulae; stochastic integration; stochastic partial differential equations; stochastic wave equation", } @Article{Holmes:2008:CLT, author = "Mark Holmes", title = "Convergence of Lattice Trees to Super-{Brownian} Motion above the Critical Dimension", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "23:671--23:755", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-499", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/499", abstract = "We use the lace expansion to prove asymptotic formulae for the Fourier transforms of the $r$-point functions for a spread-out model of critically weighted lattice trees on the $d$-dimensional integer lattice for $ d > 8$. A lattice tree containing the origin defines a sequence of measures on the lattice, and the statistical mechanics literature gives rise to a natural probability measure on the collection of such lattice trees. Under this probability measure, our results, together with the appropriate limiting behaviour for the survival probability, imply convergence to super-Brownian excursion in the sense of finite-dimensional distributions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "lace expansion.; Lattice trees; super-Brownian motion", } @Article{Roellin:2008:SCB, author = "Adrian Roellin", title = "Symmetric and centered binomial approximation of sums of locally dependent random variables", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "24:756--24:776", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-503", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/503", abstract = "Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric binomial distribution, serving as a natural alternative to the normal distribution in discrete settings. The bounds are given with respect to the total variation and a local limit metric. Under appropriate smoothness properties of the summands, the same order of accuracy as in the Berry--Ess{\'e}en Theorem is achieved. The approximation of the total number of points of a point processes is also considered. The results are applied to the exceedances of the $r$-scans process and to the Mat{\'e}rn hardcore point process type I to obtain explicit bounds with respect to the two metrics.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "binomial distribution; local dependence; Stein's method; total variation metric", } @Article{Champagnat:2008:LTC, author = "Nicolas Champagnat and Sylvie Roelly", title = "Limit theorems for conditioned multitype {Dawson--Watanabe} processes and {Feller} diffusions", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "25:777--25:810", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-504", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/504", abstract = "A multitype Dawson--Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process-the conditioned multitype Feller branching diffusion-are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "conditioned Dawson--Watanabe process; conditioned Feller diffusion; critical and subcritical Dawson--Watanabe process; long time behavior.; multitype measure-valued branching processes; remote survival", } @Article{Basdevant:2008:RGT, author = "Anne-Laure Basdevant and Arvind Singh", title = "Rate of growth of a transient cookie random walk", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "26:811--26:851", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-498", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/498", abstract = "We consider a one-dimensional transient cookie random walk. It is known from a previous paper (BS2008) that a cookie random walk $ (X_n) $ has positive or zero speed according to some positive parameter $ \alpha > 1 $ or $ \leq 1 $. In this article, we give the exact rate of growth of $ X_n $ in the zero speed regime, namely: for $ 0 < \alpha < 1 $, $ X_n / n^{(? + 1) / 2} $ converges in law to a Mittag-Leffler distribution whereas for $ \alpha = 1 $, $ X_n(\log n) / n $ converges in probability to some positive constant.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "branching process with migration; cookie or multi-excited random walk; Rates of transience", } @Article{Petrou:2008:MCL, author = "Evangelia Petrou", title = "{Malliavin} Calculus in {L{\'e}vy} spaces and Applications to Finance", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "27:852--27:879", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-502", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/502", abstract = "The main goal of this paper is to generalize the results of Fournie et al. [7] for markets generated by L{\'e}vy processes. For this reason we extend the theory of Malliavin calculus to provide the tools that are necessary for the calculation of the sensitivities, such as differentiability results for the solution of a stochastic differential equation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Windisch:2008:LCV, author = "David Windisch", title = "Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "28:880--28:897", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-506", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/506", abstract = "This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus $ ({\mathbb Z} / N{\mathbb Z})^d $ up to time $ u N^d $ in high dimension $d$. If $ u > 0$ is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length $ c_0 \log N$ for some constant $ c_0 > 0$, and this component occupies a non-degenerate fraction of the total volume as $N$ tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant $ c_0 > 0$ is crucial in the definition of the giant component.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "discrete torus; Giant component; random walk; vacant set", } @Article{Boufoussi:2008:PPC, author = "Brahim Boufoussi and Marco Dozzi and Raby Guerbaz", title = "Path properties of a class of locally asymptotically self similar processes", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "29:898--29:921", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-505", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/505", abstract = "Various paths properties of a stochastic process are obtained under mild conditions which allow for the integrability of the characteristic function of its increments and for the dependence among them. The main assumption is closely related to the notion of local asymptotic self-similarity. New results are obtained for the class of multifractional random processes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Hausdorff dimension, level sets, local asymptotic self-similarity, local non-determinism, local times", } @Article{Reynolds:2008:DRS, author = "David Reynolds and John Appleby", title = "Decay Rates of Solutions of Linear Stochastic {Volterra} Equations", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "30:922--30:943", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-507", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/507", abstract = "The paper studies the exponential and non--exponential convergence rate to zero of solutions of scalar linear convolution It{\^o}-Volterra equations in which the noise intensity depends linearly on the current state. By exploiting the positivity of the solution, various upper and lower bounds in first mean and almost sure sense are obtained, including Liapunov exponents.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "almost sure exponential asymptotic stability, Liapunov exponent, subexponential distribution, subexponential function, Volterra equations, It{\^o}-Volterra equations", } @Article{Menshikov:2008:URR, author = "Mikhail Menshikov and Stanislav Volkov", title = "Urn-related random walk with drift $ \rho x^\alpha / t^\beta $", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "31:944--31:960", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-508", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/508", abstract = "We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "martingales; Random walks; urn models", } @Article{Kulik:2008:SEV, author = "Rafal Kulik", title = "Sums of extreme values of subordinated long-range dependent sequences: moving averages with finite variance", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "32:961--32:979", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-510", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/510", abstract = "In this paper we study the limiting behavior of sums of extreme values of long range dependent sequences defined as functionals of linear processes with finite variance. If the number of extremes in a sum is large enough, we obtain asymptotic normality, however, the scaling factor is relatively bigger than in the i.i.d case, meaning that the maximal terms have relatively smaller contribution to the whole sum. Also, it is possible for a particular choice of a model, that the scaling need not to depend on the tail index of the underlying marginal distribution, as it is well-known to be so in the i.i.d. situation. Furthermore, subordination may change the asymptotic properties of sums of extremes.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "sample quantiles, linear processes, empirical processes, long range dependence, sums of extremes, trimmed sums", } @Article{Broman:2008:LLC, author = "Erik Broman and Federico Camia", title = "Large-{$N$} Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in {Mandelbrot}'s Fractal Percolation Process", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "33:980--33:999", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-511", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/511", abstract = "We study Mandelbrot's percolation process in dimension $ d \geq 2 $. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $ [0, 1]^d $ in $ N^d $ subcubes, and independently retaining or discarding each subcube with probability $p$ or $ 1 - p$ respectively. This step is then repeated within the retained subcubes at all scales. As $p$ is varied, there is a percolation phase transition in terms of paths for all $ d \geq 2$, and in terms of $ (d - 1)$-dimensional ``sheets'' for all $ d \geq 3$.\par For any $ d \geq 2$, we consider the random fractal set produced at the path-percolation critical value $ p_c(N, d)$, and show that the probability that it contains a path connecting two opposite faces of the cube $ [0, 1]^d$ tends to one as $ N \to \infty $. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of $p$, at $ p_c(N, d)$ for all $N$ sufficiently large. This had previously been proved only for $ d = 2$ (for any $ N \geq 2$). For $ d \geq 3$, we prove analogous results for sheet-percolation.\par In dimension two, Chayes and Chayes proved that $ p_c(N, 2)$ converges, as $ N \to \infty $, to the critical density $ p_c$ of site percolation on the square lattice. Assuming the existence of the correlation length exponent $ \nu $ for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that $ p_c(N, 2) - p_c = (\frac {1}{N})^{1 / \nu + o(1)}$ as $ N \to \infty $, showing an interesting relation with near-critical percolation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "critical probability; crossing probability; enhancement/diminishment percolation; Fractal percolation; near-critical percolation", } @Article{Adamczak:2008:TIS, author = "Radoslaw Adamczak", title = "A tail inequality for suprema of unbounded empirical processes with applications to {Markov} chains", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "34:1000--34:1034", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-521", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/521", abstract = "We present a tail inequality for suprema of empirical processes generated by variables with finite $ \psi_\alpha $ norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "concentration inequalities, empirical processes, Markov chains", } @Article{Matoussi:2008:SSS, author = "Anis Matoussi and Mingyu Xu", title = "{Sobolev} solution for semilinear {PDE} with obstacle under monotonicity condition", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "35:1035--35:1067", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-522", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/522", abstract = "We prove the existence and uniqueness of Sobolev solution of a semilinear PDE's and PDE's with obstacle under monotonicity condition. Moreover we give the probabilistic interpretation of the solutions in term of Backward SDE and reflected Backward SDE respectively", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Backward stochastic differential equation, Reflected backward stochastic differential equation, monotonicity condition, Stochastic flow, partial differential equation with obstacle", } @Article{DeBlassie:2008:EPB, author = "Dante DeBlassie", title = "The Exit Place of {Brownian} Motion in the Complement of a Horn", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "36:1068--36:1095", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-524", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/524", abstract = "Consider the domain lying outside a horn. We determine asymptotics of the logarithm of the chance that Brownian motion in the domain has a large exit place. For a certain class of horns, the behavior is given explicitly in terms of the geometry of the domain. We show that for some horns the behavior depends on the dimension, whereas for other horns, it does not. Analytically, the result is equivalent to estimating the harmonic measure of the part of the domain lying outside a cylinder with large diameter.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Horn-shaped domain, $h$-transform, Feynman--Kac representation, exit place of Brownian motion, harmonic measure", } @Article{Zambotti:2008:CEB, author = "Lorenzo Zambotti", title = "A conservative evolution of the {Brownian} excursion", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "37:1096--37:1119", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-525", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/525", abstract = "We consider the problem of conditioning the Brownian excursion to have a fixed time average over the interval [0, 1] and we study an associated stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian excursion; Brownian meander; singular conditioning; Stochastic partial differential equations with reflection", } @Article{Baudoin:2008:SSF, author = "Fabrice Baudoin and Laure Coutin", title = "Self-similarity and fractional {Brownian} motion on {Lie} groups", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "38:1120--38:1139", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-530", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/530", abstract = "The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are characterized.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Fractional Brownian motion, Lie group", } @Article{Basse:2008:GMA, author = "Andreas Basse", title = "{Gaussian} Moving Averages and Semimartingales", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "39:1140--39:1165", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-526", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/526", abstract = "In the present paper we study moving averages (also known as stochastic convolutions) driven by a Wiener process and with a deterministic kernel. Necessary and sufficient conditions on the kernel are provided for the moving average to be a semimartingale in its natural filtration. Our results are constructive - meaning that they provide a simple method to obtain kernels for which the moving average is a semimartingale or a Wiener process. Several examples are considered. In the last part of the paper we study general Gaussian processes with stationary increments. We provide necessary and sufficient conditions on spectral measure for the process to be a semimartingale.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Gaussian processes; moving averages; non-canonical representations; semimartingales; stationary processes; stochastic convolutions", } @Article{Alberts:2008:HDS, author = "Tom Alberts and Scott Sheffield", title = "{Hausdorff} Dimension of the {SLE} Curve Intersected with the Real Line", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "40:1166--40:1188", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-515", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/515", abstract = "We establish an upper bound on the asymptotic probability of an $ S L E(\kappa) $ curve hitting two small intervals on the real line as the interval width goes to zero, for the range $ 4 < \kappa < 8 $. As a consequence we are able to prove that the random set of points in $R$ hit by the curve has Hausdorff dimension $ 2 - 8 / \kappa $, almost surely.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Hausdorff dimension; SLE; Two-point hitting probability", } @Article{Muller:2008:CTM, author = "Sebastian M{\"u}ller", title = "A criterion for transience of multidimensional branching random walk in random environment", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "41:1189--41:1202", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-517", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/517", abstract = "We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that Condition L of Comets and Popov [3] is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Branching Markov chains; random environment, spectral radius; recurrence; transience", } @Article{Cox:2008:CMW, author = "Alexander Cox and Jan Obloj", title = "Classes of measures which can be embedded in the Simple Symmetric Random Walk", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "42:1203--42:1228", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-516", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/516", abstract = "We characterize the possible distributions of a stopped simple symmetric random walk $ X_\tau $, where $ \tau $ is a stopping time relative to the natural filtration of $ (X_n) $. We prove that any probability measure on $ \mathbb {Z} $ can be achieved as the law of $ X_\tau $ where $ \tau $ is a minimal stopping time, but the set of measures obtained under the further assumption that $ (X_{n \land \tau } \colon n \geq 0) $ is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on $ \mathbb {Z} $. This is in sharp contrast to the well-studied Brownian motion setting. We also investigate the discrete counterparts of the Chacon-Walsh (1976) and Azema-Yor (1979) embeddings and show that they lead to yet smaller sets of achievable measures. Finally, we solve explicitly the Skorokhod embedding problem constructing, for a given measure $ \mu $, a minimal stopping time $ \tau $ which embeds $ \mu $ and which further is uniformly integrable whenever a uniformly integrable embedding of $ \mu $ exists.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Azema-Yor stopping time; Chacon-Walsh stopping time; fractal; iterated function system; minimal stopping time; random walk; self-similar set; Skorokhod embedding problem; uniform integrability", } @Article{Nourdin:2008:WPV, author = "Ivan Nourdin and Giovanni Peccati", title = "Weighted power variations of iterated {Brownian} motion", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "43:1229--43:1256", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-534", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/534", abstract = "We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the limiting objects can always be expressed in terms of three independent Brownian motions $ X, Y $ and $B$, as well as of the local times of $Y$. In particular, our results involve ''weighted'' versions of Kesten and Spitzer's Brownian motion in random scenery. Our findings extend the theory initiated by Khoshnevisan and Lewis (1999), and should be compared with the recent result by Nourdin and R{\'e}veillac (2008), concerning the weighted power variations of fractional Brownian motion with Hurst index $ H = 1 / 4$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion; Brownian motion in random scenery; Iterated Brownian motion; Limit theorems; Weighted power variations", } @Article{Gibson:2008:MSV, author = "Lee Gibson", title = "The mass of sites visited by a random walk on an infinite graph", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "44:1257--44:1282", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-531", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/531", abstract = "We determine the log-asymptotic decay rate of the negative exponential moments of the mass of sites visited by a random walk on an infinite graph which satisfies a two-sided sub-Gaussian estimate on its transition kernel. This provides a new method of proof of the correct decay rate for Cayley graphs of finitely generated groups with polynomial volume growth. This method also extend known results by determining this decay rate for certain graphs with fractal-like structure or with non-Alfors regular volume growth functions.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random walk, infinite graph, visited sites, asymptotic decay rates, polynomial volume growth, Cayley graph, fractal graph, Alfors regular", } @Article{Davies:2008:SAN, author = "Ian Davies", title = "Semiclassical Analysis and a New Result for {Poisson--L{\'e}vy} Excursion Measures", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "45:1283--45:1306", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-513", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/513", abstract = "The Poisson--L{\'e}vy excursion measure for the diffusion process with small noise satisfying the It{\^o} equation\par $$ d X^{\varepsilon } = b(X^{\varepsilon }(t))d t + \sqrt \varepsilon \, d B(t) $$ is studied and the asymptotic behaviour in $ \varepsilon $ is investigated. The leading order term is obtained exactly and it is shown that at an equilibrium point there are only two possible forms for this term --- Levy or Hawkes--Truman. We also compute the next to leading order.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "excursion measures, asymptotic expansions", } @Article{Eichelsbacher:2008:ORW, author = "Peter Eichelsbacher and Wolfgang K{\"o}nig", title = "Ordered Random Walks", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "46:1307--46:1336", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-539", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/539", abstract = "We construct the conditional version of $k$ independent and identically distributed random walks on $R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Doob h-transform; Dyson's Brownian motions; fluctuation theory.; non-colliding random walks; non-intersecting random processes; Vandermonde determinant", } @Article{Kulske:2008:PMG, author = "Christof K{\"u}lske and Alex Opoku", title = "The posterior metric and the goodness of {Gibbsianness} for transforms of {Gibbs} measures", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "47:1307--47:1344", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-560", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/560", abstract = "We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models subjected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. Assuming no a priori metric on the local state spaces but only a measurable structure, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the $ (q - 1)$-dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric. In another application we prove the preservation of Gibbsianness for sufficiently fine local coarse-grainings when the Hamiltonian satisfies a Lipschitz property", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "phase transitions; posterior metric; specification; Time-evolved Gibbs measures, non-Gibbsian measures: Dobrushin uniqueness", } @Article{Collet:2008:RPS, author = "Pierre Collet and Antonio Galves and Florencia Leonardi", title = "Random perturbations of stochastic processes with unbounded variable length memory", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "48:1345--48:1361", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-538", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/538", abstract = "We consider binary infinite order stochastic chains perturbed by a random noise. This means that at each time step, the value assumed by the chain can be randomly and independently flipped with a small fixed probability. We show that the transition probabilities of the perturbed chain are uniformly close to the corresponding transition probabilities of the original chain. As a consequence, in the case of stochastic chains with unbounded but otherwise finite variable length memory, we show that it is possible to recover the context tree of the original chain, using a suitable version of the algorithm Context, provided that the noise is small enough.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "chains of infinite order, variable length Markov chains, chains with unbounded variable length memory, random perturbations, algorithm Context, context trees", } @Article{Bonaccorsi:2008:SFN, author = "Stefano Bonaccorsi and Carlo Marinelli and Giacomo Ziglio", title = "Stochastic {FitzHugh--Nagumo} equations on networks with impulsive noise", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "49:1362--49:1379", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-532", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/532", abstract = "We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh--Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary value problem as a stochastic evolution equation in infinite dimensions with a power-type nonlinearity, driven by an additive L{\'e}vy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic PDEs, FitzHugh--Nagumo equation, L{\'e}vy processes, maximal monotone operators", } @Article{Borodin:2008:LTA, author = "Alexei Borodin and Patrik Ferrari", title = "Large time asymptotics of growth models on space-like paths {I}: {PushASEP}", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "50:1380--50:1418", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-541", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/541", abstract = "We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stochastic growth, KPZ, determinantal processes, Airy processes", } @Article{Croydon:2008:RWG, author = "David Croydon and Takashi Kumagai", title = "Random walks on {Galton--Watson} trees with infinite variance offspring distribution conditioned to survive", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "51:1419--51:1441", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-536", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/536", abstract = "We establish a variety of properties of the discrete time simple random walk on a Galton--Watson tree conditioned to survive when the offspring distribution, $Z$ say, is in the domain of attraction of a stable law with index $ \alpha \in (1, 2]$. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is $ 2 \alpha / (2 \alpha - 1)$. Furthermore, we demonstrate that when $ \alpha \in (1, 2)$ there are logarithmic fluctuations in the quenched transition density of the simple random walk, which contrasts with the log-logarithmic fluctuations seen when $ \alpha = 2$. In the course of our arguments, we obtain tail bounds for the distribution of the $n$ th generation size of a Galton--Watson branching process with offspring distribution $Z$ conditioned to survive, as well as tail bounds for the distribution of the total number of individuals born up to the $n$ th generation, that are uniform in $n$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "branching process; random walk; stable distribution; transition density", } @Article{Schweinsberg:2008:WM, author = "Jason Schweinsberg", title = "Waiting for $m$ mutations", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "52:1442--52:1478", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-540", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/540", abstract = "We consider a model of a population of fixed size $N$ in which each individual gets replaced at rate one and each individual experiences a mutation at rate $ \mu $. We calculate the asymptotic distribution of the time that it takes before there is an individual in the population with $m$ mutations. Several different behaviors are possible, depending on how ?? changes with $N$. These results have applications to the problem of determining the waiting time for regulatory sequences to appear and to models of cancer development.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Moran model; mutations; population genetics; Waiting times", } @Article{Voss:2008:LDO, author = "Jochen Voss", title = "Large Deviations for One Dimensional Diffusions with a Strong Drift", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "53:1479--53:1528", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-564", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/564", abstract = "We derive a large deviation principle which describes the behaviour of a diffusion process with additive noise under the influence of a strong drift. Our main result is a large deviation theorem for the distribution of the end-point of a one-dimensional diffusion with drift $ \theta b $ where $b$ is a drift function and $ \theta $ a real number, when $ \theta $ converges to $ \infty $. It transpires that the problem is governed by a rate function which consists of two parts: one contribution comes from the Freidlin--Wentzell theorem whereas a second term reflects the cost for a Brownian motion to stay near a equilibrium point of the drift over long periods of time.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "diffusion processes; large deviations; stochastic differential equations", } @Article{Confortola:2008:QBR, author = "Fulvia Confortola and Philippe Briand", title = "Quadratic {BSDEs} with Random Terminal Time and Elliptic {PDEs} in Infinite Dimension", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "54:1529--54:1561", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-514", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/514", abstract = "In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator $ F(t, Y, Z) $ has a quadratic growth in $Z$. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces. Finally we show an application to a control problem.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "elliptic PDEs; optimal stochastic control; Quadratic BSDEs", } @Article{Nolin:2008:NCP, author = "Pierre Nolin", title = "Near-critical percolation in two dimensions", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "55:1562--55:1623", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-565", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/565", abstract = "We give a self-contained and detailed presentation of Kesten's results that allow to relate critical and near-critical percolation on the triangular lattice. They constitute an important step in the derivation of the exponents describing the near-critical behavior of this model. For future use and reference, we also show how these results can be obtained in more general situations, and we state some new consequences.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "arm events; critical exponents; near-critical percolation", } @Article{Albenque:2008:SFI, author = "Marie Albenque and Jean-Fran{\c{c}}ois Marckert", title = "Some families of increasing planar maps", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "56:1624--56:1671", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-563", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/563", abstract = "Stack-triangulations appear as natural objects when one wants to define some families of increasing triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $ 2 n $ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $ n^{1 / 2} $, they converge for the Gromov--Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $ (6 / 11) \log n $ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "stackmaps, triangulations, Gromov--Hausdorff convergence, continuum random tree", } @Article{Kyprianou:2008:SCC, author = "Andreas Kyprianou and Victor Rivero", title = "Special, conjugate and complete scale functions for spectrally negative {L{\'e}vy} processes", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "57:1672--57:1701", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-567", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/567", abstract = "Following from recent developments in Hubalek and Kyprianou [28], the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative L{\'e}vy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener--Hopf factorization for spectrally negative L{\'e}vy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Potential theory for subordinators, Scale functions, Special subordinators, Spectrally negative L{\'e}vy processes", } @Article{Lyons:2008:EUS, author = "Russell Lyons and Benjamin Morris and Oded Schramm", title = "Ends in Uniform Spanning Forests", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "58:1702--58:1725", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-566", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/566", abstract = "It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Cayley graphs.; Spanning trees", } @Article{Gayrard:2008:EPT, author = "V{\'e}ronique Gayrard and G{\'e}rard Ben Arous", title = "Elementary potential theory on the hypercube", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "59:1726--59:1807", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-527", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/527", abstract = "This work addresses potential theoretic questions for the standard nearest neighbor random walk on the hypercube $ \{ - 1, + 1 \}^N $. For a large class of subsets $ A \subset \{ - 1, + 1 \}^N $ we give precise estimates for the harmonic measure of $A$, the mean hitting time of $A$, and the Laplace transform of this hitting time. In particular, we give precise sufficient conditions for the harmonic measure to be asymptotically uniform, and for the hitting time to be asymptotically exponentially distributed, as $ N \rightarrow \infty $. Our approach relies on a $d$-dimensional extension of the Ehrenfest urn scheme called lumping and covers the case where $d$ is allowed to diverge with $N$ as long as $ d \leq \alpha_0 \frac {N}{\log N}$ for some constant $ 0 < \alpha_0 < 1$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random walk on hypercubes, lumping", } @Article{Bass:2008:DSD, author = "Richard Bass and Edwin Perkins", title = "Degenerate stochastic differential equations arising from catalytic branching networks", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "60:1808--60:1885", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-568", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/568", abstract = "We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. The drift and branching coefficients are only assumed to be continuous and satisfy some natural non-degeneracy conditions. We assume at most one catalyst per site as is the case for the hypercyclic equation. Here the two-dimensional case with affine drift is required in work of [DGHSS] on mean fields limits of block averages for 2-type branching models on a hierarchical group. The proofs make use of some new methods, including Cotlar's lemma to establish asymptotic orthogonality of the derivatives of an associated semigroup at different times, and a refined integration by parts technique from [DP1].", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "catalytic branching; Cotlar's lemma; degenerate diffusions; martingale problem; perturbations; resolvents; stochastic differential equations", } @Article{Piera:2008:CRR, author = "Francisco Piera and Ravi Mazumdar", title = "Comparison Results for Reflected Jump-diffusions in the Orthant with Variable Reflection Directions and Stability Applications", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "61:1886--61:1908", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-569", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/569", abstract = "We consider reflected jump-diffusions in the orthant $ R_+^n $ with time- and state-dependent drift, diffusion and jump-amplitude coefficients. Directions of reflection upon hitting boundary faces are also allow to depend on time and state. Pathwise comparison results for this class of processes are provided, as well as absolute continuity properties for their associated regulator processes responsible of keeping the respective diffusions in the orthant. An important role is played by the boundary property in that regulators do not charge times spent by the reflected diffusion at the intersection of two or more boundary faces. The comparison results are then applied to provide an ergodicity condition for the state-dependent reflection directions case.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "ergodicity.; Jump-diffusion processes; pathwise comparisons; Skorokhod maps; stability; state-dependent oblique reflections", } @Article{Veto:2008:SRR, author = "Balint Veto and Balint Toth", title = "Self-repelling random walk with directed edges on {$ \mathbb {Z} $}", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "62:1909--62:1926", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-570", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/570", abstract = "We consider a variant of self-repelling random walk on the integer lattice Z where the self-repellence is defined in terms of the local time on oriented edges. The long-time asymptotic scaling of this walk is surprisingly different from the asymptotics of the similar process with self-repellence defined in terms of local time on unoriented edges. We prove limit theorems for the local time process and for the position of the random walker. The main ingredient is a Ray--Knight-type of approach. At the end of the paper, we also present some computer simulations which show the strange scaling behaviour of the walk considered.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "random walks with long memory, self-repelling, one dimension, oriented edges, local time, Ray--Knight-theory, coupling", } @Article{Amir:2008:SSE, author = "Gideon Amir and Christopher Hoffman", title = "A special set of exceptional times for dynamical random walk on {$ Z^2 $}", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "63:1927--63:1951", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-571", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/571", abstract = "In [2] Benjamini, H{\"a}ggstr{\"o}m, Peres and Steif introduced the model of dynamical random walk on the $d$-dimensional lattice $ Z^d$. This is a continuum of random walks indexed by a time parameter $t$. They proved that for dimensions $ d = 3, 4$ there almost surely exist times $t$ such that the random walk at time $t$ visits the origin infinitely often, but for dimension 5 and up there almost surely do not exist such $t$. Hoffman showed that for dimension 2 there almost surely exists $t$ such that the random walk at time $t$ visits the origin only finitely many times [5]. We refine the results of [5] for dynamical random walk on $ Z^2$, showing that with probability one the are times when the origin is visited only a finite number of times while other points are visited infinitely often.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Dynamical Random Walks, Dynamical Sensativity; Random Walks", } @Article{Kosygina:2008:PNE, author = "Elena Kosygina and Martin Zerner", title = "Positively and negatively excited random walks on integers, with branching processes", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "64:1952--64:1979", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-572", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/572", abstract = "We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Central limit theorem; excited random walk; law of large numbers; positive and negative cookies; recurrence; renewal structure; transience", } @Article{Bianchi:2008:GDN, author = "Alessandra Bianchi", title = "{Glauber} dynamics on nonamenable graphs: boundary conditions and mixing time", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "65:1980--65:2012", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-574", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/574", abstract = "We study the stochastic Ising model on finite graphs with n vertices and bounded degree and analyze the effect of boundary conditions on the mixing time. We show that for all low enough temperatures, the spectral gap of the dynamics with (+)-boundary condition on a class of nonamenable graphs, is strictly positive uniformly in n. This implies that the mixing time grows at most linearly in n. The class of graphs we consider includes hyperbolic graphs with sufficiently high degree, where the best upper bound on the mixing time of the free boundary dynamics is polynomial in n, with exponent growing with the inverse temperature. In addition, we construct a graph in this class, for which the mixing time in the free boundary case is exponentially large in n. This provides a first example where the mixing time jumps from exponential to linear in n while passing from free to (+)-boundary condition. These results extend the analysis of Martinelli, Sinclair and Weitz to a wider class of nonamenable graphs.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Glauber dynamics; mixing time; nonamenable graphs; spectral gap", } @Article{Bordenave:2008:BAP, author = "Charles Bordenave", title = "On the birth-and-assassination process, with an application to scotching a rumor in a network", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "66:2014--66:2030", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-573", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/573", abstract = "We give new formulas on the total number of born particles in the stable birth-and-assassination process, and prove that it has a heavy-tailed distribution. We also establish that this process is a scaling limit of a process of rumor scotching in a network, and is related to a predator-prey dynamics.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "branching process, heavy tail phenomena, SIR epidemics", } @Article{Neuenkirch:2008:DED, author = "Andreas Neuenkirch and Ivan Nourdin and Samy Tindel", title = "Delay equations driven by rough paths", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "67:2031--67:2068", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-575", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/575", abstract = "In this article, we illustrate the flexibility of the algebraic integration formalism introduced in M. Gubinelli, {\em J. Funct. Anal.} {\bf 216}, 86-140, 2004, \url{http://www.ams.org/mathscinet-getitem?mr=2005k:60169} Math. Review 2005k:60169, by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter $ H > 1 / 3 $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "delay equation; fractional Brownian motion; Malliavin calculus; rough paths theory", } @Article{Hermisson:2008:PGH, author = "Joachim Hermisson and Peter Pfaffelhuber", title = "The pattern of genetic hitchhiking under recurrent mutation", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "68:2069--68:2106", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-577", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/577", abstract = "Genetic hitchhiking describes evolution at a neutral locus that is linked to a selected locus. If a beneficial allele rises to fixation at the selected locus, a characteristic polymorphism pattern (so-called selective sweep) emerges at the neutral locus. The classical model assumes that fixation of the beneficial allele occurs from a single copy of this allele that arises by mutation. However, recent theory (Pennings and Hermisson, 2006a, b) has shown that recurrent beneficial mutation at biologically realistic rates can lead to markedly different polymorphism patterns, so-called soft selective sweeps. We extend an approach that has recently been developed for the classical hitchhiking model (Schweinsberg and Durrett, 2005; Etheridge et al., 2006) to study the recurrent mutation scenario. We show that the genealogy at the neutral locus can be approximated (to leading orders in the selection strength) by a marked Yule process with immigration. Using this formalism, we derive an improved analytical approximation for the expected heterozygosity at the neutral locus at the time of fixation of the beneficial allele.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Selective sweep, genetic hitchhiking, soft selective sweep, diffusion approximation, Yule process, random background", } @Article{Arguin:2008:CPS, author = "Louis-Pierre Arguin", title = "Competing Particle Systems and the {Ghirlanda--Guerra} Identities", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "69:2101--69:2117", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-579", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/579", abstract = "Competing particle systems are point processes on the real line whose configurations $X$ can be ordered decreasingly and evolve by increments which are functions of correlated Gaussian variables. The correlations are intrinsic to the points and quantified by a matrix $ Q = \{ q_{ij} \} $. Quasi-stationary systems are those for which the law of $ (X, Q)$ is invariant under the evolution up to translation of $X$. It was conjectured by Aizenman and co-authors that the matrix $Q$ of robustly quasi-stationary systems must exhibit a hierarchical structure. This was established recently, up to a natural decomposition of the system, whenever the set $ S_Q$ of values assumed by $ q_{ij}$ is finite. In this paper, we study the general case where $ S_Q$ may be infinite. Using the past increments of the evolution, we show that the law of robustly quasi-stationary systems must obey the Ghirlanda--Guerra identities, which first appear in the study of spin glass models. This provides strong evidence that the above conjecture also holds in the general case. In addition, it yields an alternative proof of a theorem of Ruzmaikina and Aizenman for independent increments.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Point processes, Ultrametricity, Ghirlanda--Guerra identities", } @Article{Garet:2008:FPC, author = "Olivier Garet and R{\'e}gine Marchand", title = "First-passage competition with different speeds: positive density for both species is impossible", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "70:2118--70:2159", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-581", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/581", abstract = "Consider two epidemics whose expansions on $ \mathbb {Z}^d $ are governed by two families of passage times that are distinct and stochastically comparable. We prove that when the weak infection survives, the space occupied by the strong one is almost impossible to detect. Particularly, in dimension two, we prove that one species finally occupies a set with full density, while the other one only occupies a set of null density. Furthermore, we observe the same fluctuations with respect to the asymptotic shape as for the weak infection evolving alone. By the way, we extend the H{\"a}ggstr{\"o}m-Pemantle non-coexistence result ``except perhaps for a denumerable set'' to families of stochastically comparable passage times indexed by a continuous parameter.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "coexistence; competition; first-passage percolation; moderate deviations; random growth", } @Article{Athreya:2008:RDT, author = "Siva Athreya and Rahul Roy and Anish Sarkar", title = "Random directed trees and forest --- drainage networks with dependence", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "71:2160--71:2189", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-580", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/580", abstract = "Consider the $d$-dimensional lattice $ \mathbb Z^d$ where each vertex is `open' or `closed' with probability $p$ or $ 1 - p$ respectively. An open vertex $v$ is connected by an edge to the closest open vertex $ w$ in the $ 45^\circ $ (downward) light cone generated at $v$. In case of non-uniqueness of such a vertex $w$, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for $ d = 2$ and $3$ and it is an infinite collection of distinct trees for $ d \geq 4$. In addition, for any dimension, we show that there is no bi-infinite path in the tree.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Random Graph, Random Oriented Trees, Random Walk", } @Article{Heunis:2008:ICN, author = "Andrew Heunis and Vladimir Lucic", title = "On the Innovations Conjecture of Nonlinear Filtering with Dependent Data", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "72:2190--72:2216", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-585", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/585", abstract = "We establish the innovations conjecture for a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The approach uses only elementary stochastic analysis, together with a variant due to J. M. C. Clark of a theorem of Yamada and Watanabe on pathwise-uniqueness and strong solutions of stochastic differential equations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "innovations conjecture; nonlinear filter; pathwise-uniqueness", } @Article{Faggionato:2008:RWE, author = "Alessandra Faggionato", title = "Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "73:2217--73:2247", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-591", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/591", abstract = "We consider a stationary and ergodic random field $ \{ \omega (b) \colon b \in \mathbb {E}_d \} $ parameterized by the family of bonds in $ \mathbb {Z}^d $, $ d \geq 2 $. The random variable $ \omega (b) $ is thought of as the conductance of bond $b$ and it ranges in a finite interval $ [0, c_0]$. Assuming that the set of bonds with positive conductance has a unique infinite cluster $ \mathcal {C}(\omega)$, we prove homogenization results for the random walk among random conductances on $ \mathcal {C}(\omega)$. As a byproduct, applying the general criterion of Faggionato (2007) leading to the hydrodynamic limit of exclusion processes with bond--dependent transition rates, for almost all realizations of the environment we prove the hydrodynamic limit of simple exclusion processes among random conductances on $ \mathcal {C}(\omega)$. The hydrodynamic equation is given by a heat equation whose diffusion matrix does not depend on the environment. We do not require any ellipticity condition. As special case, $ \mathcal {C}(\omega)$ can be the infinite cluster of supercritical Bernoulli bond percolation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "bond percolation; disordered system; exclusion process; homogenization; random walk in random environment", } @Article{Mueller:2008:RDS, author = "Carl Mueller and David Nualart", title = "Regularity of the density for the stochastic heat equation", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "74:2248--74:2258", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-589", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/589", abstract = "We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "heat equation, white noise, Malliavin calculus, stochastic partial differential equations", } @Article{Zemlys:2008:HFS, author = "Vaidotas Zemlys", title = "A {H{\"o}lderian} {FCLT} for some multiparameter summation process of independent non-identically distributed random variables", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "75:2259--75:2282", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-590", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/590", abstract = "We introduce a new construction of a summation process based on the collection of rectangular subsets of unit d-dimensional cube for a triangular array of independent non-identically distributed variables with d-dimensional index, using the non-uniform grid adapted to the variances of the variables. We investigate its convergence in distribution in some Holder spaces. It turns out that for dimensions greater than 2, the limiting process is not necessarily the standard Brownian sheet. This contrasts with a classical result of Prokhorov for the one-dimensional case.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian sheet, functional central limit theorem, H{\"o}lder space, invariance principle, triangular array, summation process.", } @Article{Drewitz:2008:LEO, author = "Alexander Drewitz", title = "{Lyapunov} exponents for the one-dimensional parabolic {Anderson} model with drift", journal = j-ELECTRON-J-PROBAB, volume = "13", pages = "76:2283--76:2336", year = "2008", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v13-586", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/586", abstract = "We consider the solution to the one-dimensional parabolic Anderson model with homogeneous initial condition, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for all positive real $p$. These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p$ tends to 0. Furthermore, we show that the solution is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman--Kac representation of the solution under the corresponding Gibbs measure. In our context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears as the drift parameter or diffusion constant increase, respectively.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Parabolic Anderson model, Lyapunov exponents, intermittency, large deviations", } @Article{Hambly:2009:PHI, author = "Ben Hambly and Martin Barlow", title = "Parabolic {Harnack} inequality and local limit theorem for percolation clusters", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "1:1--1:26", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-587", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/587", abstract = "We consider the random walk on supercritical percolation clusters in $ \mathbb {Z}^d $. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Harnack inequality; local limit theorem; Percolation; random walk", } @Article{Douc:2009:FIC, author = "Randal Douc and Eric Moulines and Yaacov Ritov", title = "Forgetting of the initial condition for the filter in general state-space hidden {Markov} chain: a coupling approach", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "2:27--2:49", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-593", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/593", abstract = "We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "hidden Markov chain; non-linear filtering, coupling; stability", } @Article{Atar:2009:ETG, author = "Rami Atar and Siva Athreya and Zhen-Qing Chen", title = "Exit Time, Green Function and Semilinear Elliptic Equations", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "3:50--3:71", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-597", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/597", abstract = "Let $D$ be a bounded Lipschitz domain in $ R^n$ with $ n \geq 2$ and $ \tau_D$ be the first exit time from $D$ by Brownian motion on $ R^n$. In the first part of this paper, we are concerned with sharp estimates on the expected exit time $ E_x [\tau_D]$. We show that if $D$ satisfies a uniform interior cone condition with angle $ \theta \in (\cos^{-1}(1 / \sqrt {n}), \pi)$, then $ c_1 \varphi_1 (x) \leq E_x [\tau_D] \leq c_2 \varphi_1 (x)$ on $D$. Here $ \varphi_1$ is the first positive eigenfunction for the Dirichlet Laplacian on $D$. The above result is sharp as we show that if $D$ is a truncated circular cone with angle $ \theta < \cos^{-1}(1 / \sqrt {n})$, then the upper bound for $ E_x [\tau_D]$ fails. These results are then used in the second part of this paper to investigate whether positive solutions of the semilinear equation $ \Delta u = u^p$ in $ D, $ $ p \in R$, that vanish on an open subset $ \Gamma \subset \partial D$ decay at the same rate as $ \varphi_1$ on $ \Gamma $.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "boundary Harnack principle; Brownian motion; Dirichlet Laplacian; exit time; Feynman--Kac transform; Green function estimates; ground state; Lipschitz domain; Schauder's fixed point theorem; semilinear elliptic equation", } @Article{Ibarrola:2009:FTR, author = "Ricardo V{\'e}lez Ibarrola and Tomas Prieto-Rumeau", title = "{De Finetti}'s-type results for some families of non identically distributed random variables", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "4:72--4:86", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-602", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/602", abstract = "We consider random selection processes of weighted elements in an arbitrary set. Their conditional distributions are shown to be a generalization of the hypergeometric distribution, while the marginal distributions can always be chosen as generalized binomial distributions. Then we propose sufficient conditions on the weight function ensuring that the marginal distributions are necessarily of the generalized binomial form. In these cases, the corresponding indicator random variables are conditionally independent (as in the classical De Finetti theorem) though they are neither exchangeable nor identically distributed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "De Finetti theorem; exchangeability; random assignment processes", } @Article{Janson:2009:PRG, author = "Svante Janson", title = "On percolation in random graphs with given vertex degrees", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "5:86--5:118", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-603", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/603", abstract = "We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "bootstrap percolation; giant component; k-core; random graph", } @Article{Sega:2009:LRC, author = "Gregor Sega", title = "Large-range constant threshold growth model in one dimension", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "6:119--6:138", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-598", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/598", abstract = "We study a one dimensional constant threshold model in continuous time. Its dynamics have two parameters, the range $n$ and the threshold $v$. An unoccupied site $x$ becomes occupied at rate 1 as soon as there are at least $v$ occupied sites in $ [x - n, x + n]$. As n goes to infinity and $v$ is kept fixed, the dynamics can be approximated by a continuous space version, which has an explicit invariant measure at the front. This allows us to prove that the speed of propagation is asymptoticaly $ n^2 / 2 v$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "asymptotic propagation velocity; growth model; invariant distribution", } @Article{Weiss:2009:EBS, author = "Alexander Weiss", title = "Escaping the {Brownian} stalkers", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "7:139--7:160", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-594", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/594", abstract = "We propose a simple model for the behaviour of longterm investors on a stock market. It consists of three particles that represent the stock's current price and the buyers', respectively sellers', opinion about the right trading price. As time evolves, both groups of traders update their opinions with respect to the current price. The speed of updating is controlled by a parameter; the price process is described by a geometric Brownian motion. We consider the market's stability in terms of the distance between the buyers' and sellers' opinion, and prove that the distance process is recurrent/transient in dependence on the parameter.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "financial markets; market stability; recurrence; stochastic dynamics; transience", } @Article{Bovier:2009:ASS, author = "Anton Bovier and Anton Klimovsky", title = "The {Aizenman--Sims--Starr} and {Guerras} schemes for the {SK} model with multidimensional spins", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "8:161--8:241", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-611", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/611", abstract = "We prove upper and lower bounds on the free energy of the Sherrington--Kirkpatrick model with multidimensional spins in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional. We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle's probability cascades. For this purpose, an abstract quenched large deviations principle of the G{\"a}rtner-Ellis type is obtained. We derive Talagrand's representation of Guerra's remainder term for the Sherrington--Kirkpatrick model with multidimensional spins. The derivation is based on well-known properties of Ruelle's probability cascades and the Bolthausen--Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which shed some light on the original convexity problem of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of a priori estimates.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Sherrington--Kirkpatrick model, multidimensional spins, quenched large deviations, concentration of measure, Gaussian spins, convexity, Parisi functional, Parisi formula", } @Article{Taylor:2009:CPS, author = "Jesse Taylor and Amandine V{\'e}ber", title = "Coalescent processes in subdivided populations subject to recurrent mass extinctions", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "9:242--9:288", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-595", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/595", abstract = "We investigate the infinitely many demes limit of the genealogy of a sample of individuals from a subdivided population that experiences sporadic mass extinction events. By exploiting a separation of time scales that occurs within a class of structured population models generalizing Wright's island model, we show that as the number of demes tends to infinity, the limiting form of the genealogy can be described in terms of the alternation of instantaneous scattering phases that depend mainly on local demographic processes, and extended collecting phases that are dominated by global processes. When extinction and recolonization events are local, the genealogy is described by Kingman's coalescent, and the scattering phase influences only the overall rate of the process. In contrast, if the demes left vacant by a mass extinction event are recolonized by individuals emerging from a small number of demes, then the limiting genealogy is a coalescent process with simultaneous multiple mergers (a $ \Xi $-coalescent). In this case, the details of the within-deme population dynamics influence not only the overall rate of the coalescent process, but also the statistics of the complex mergers that can occur within sample genealogies. These results suggest that the combined effects of geography and disturbance could play an important role in producing the unusual patterns of genetic variation documented in some marine organisms with high fecundity.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "disturbance; extinction/recolonization; genealogy; metapopulation; population genetics; separation of time scales; Xi-coalescent", } @Article{Alsmeyer:2009:LTM, author = "Gerold Alsmeyer and Alex Iksanov", title = "A Log-Type Moment Result for Perpetuities and Its Application to Martingales in Supercritical Branching Random Walks", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "10:289--10:313", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-596", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/596", abstract = "Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons. As a by-product, necessary and sufficient conditions for uniform integrability of these martingales are provided in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "branching random walk; martingale; moments; perpetuity", } @Article{Foondun:2009:HKE, author = "Mohammud Foondun", title = "Heat kernel estimates and {Harnack} inequalities for some {Dirichlet} forms with non-local part", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "11:314--11:340", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-604", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/604", abstract = "We consider the Dirichlet form given by\par $$ {\cal E}(f, f) = \frac {1}{2} \int_{R^d} \sum_{i, j = 1}^d a_{ij}(x) \frac {\partial f(x)}{\partial x_i} \frac {\partial f(x)}{\partial x_j} d x $$ $$ + \int_{R^d \times R^d} (f(y) - f(x))^2 J(x, y)d x d y. $$ Under the assumption that the $ {a_{ij}} $ are symmetric and uniformly elliptic and with suitable conditions on $J$, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to $ \cal E$.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Integro-differential operators. Harnack inequality. Heat kernel, Holder continuity", } @Article{Lejay:2009:RDE, author = "Antoine Lejay", title = "On rough differential equations", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "12:341--12:364", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-613", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/613", abstract = "We prove that the It{\^o} map, that is the map that gives the solution of a differential equation controlled by a rough path of finite $p$-variation with $ p \in [2, 3)$ is locally Lipschitz continuous in all its arguments and we give some sufficient conditions for global existence for non-bounded vector fields.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", } @Article{Barbour:2009:SCI, author = "A. Barbour and A. Gnedin", title = "Small counts in the infinite occupancy scheme", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "13:365--13:384", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-608", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/608", abstract = "The paper is concerned with the classical occupancy scheme in which balls are thrown independently into infinitely many boxes, with given probability of hitting each of the boxes. We establish joint normal approximation, as the number of balls goes to infinity, for the numbers of boxes containing any fixed number of balls, standardized in the natural way, assuming only that the variances of these counts all tend to infinity. The proof of this approximation is based on a de-Poissonization lemma. We then review sufficient conditions for the variances to tend to infinity. Typically, the normal approximation does not mean convergence. We show that the convergence of the full vector of counts only holds under a condition of regular variation, thus giving a complete characterization of possible limit correlation structures.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "normal approximation; occupancy problem; Poissonization; regular variation", } @Article{Gravner:2009:LBP, author = "Janko Gravner and Alexander Holroyd", title = "Local Bootstrap Percolation", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "14:385--14:399", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-607", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/607", abstract = "We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single {\em active} site at the origin, while other sites of $ \mathbb {Z}^2 $ are independently {\em occupied} with small probability $p$, otherwise {\em empty}. Subsequently, an empty site becomes active by contact with two or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability $ \exp [\alpha (p) / p]$, where $ \alpha (p)$ is between $ - \pi^2 / 9 + c \sqrt p$ and $ - \pi^2 / 9 + C \sqrt p(\log p^{-1})^3$. This corrects previous numerical predictions for the scaling of the correction term.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "bootstrap percolation; cellular automaton; crossover; finite-size scaling; metastability", } @Article{Chen:2009:NFM, author = "Bo Chen and Daniel Ford and Matthias Winkel", title = "A new family of {Markov} branching trees: the alpha-gamma model", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "15:400--15:430", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-616", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/616", abstract = "We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in size-biased order, and we study their limiting behaviour.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Alpha-gamma tree, splitting rule, sampling consistency, self-similar fragmentation, dislocation measure, continuum random tree, R-tree, Markov branching model", } @Article{Tournier:2009:IET, author = "Laurent Tournier", title = "Integrability of exit times and ballisticity for random walks in {Dirichlet} environment", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "16:431--16:451", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-609", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/609", abstract = "We consider random walks in Dirichlet random environment. Since the Dirichlet distribution is not uniformly elliptic, the annealed integrability of the exit time out of a given finite subset is a non-trivial question. In this paper we provide a simple and explicit equivalent condition for the integrability of Green functions and exit times on any finite directed graph. The proof relies on a quotienting procedure allowing for an induction argument on the cardinality of the graph. This integrability problem arises in the definition of Kalikow auxiliary random walk. Using a particular case of our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "ballisticity; Dirichlet distribution; exit time; quotient graph; random walks in random environment; reinforced random walks", } @Article{Bryc:2009:DRQ, author = "W{\l}odek Bryc and Virgil Pierce", title = "Duality of real and quaternionic random matrices", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "17:452--17:476", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-606", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/606", abstract = "We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of random matrices the result gives the duality between moments of these families and the corresponding real Wigner and Wishart families.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Gaussian Symplectic Ensemble, quaternion Wishart, moments, Mobius graphs, Euler characteristic", } @Article{Bahlali:2009:HSP, author = "Khaled Bahlali and A. Elouaflin and Etienne Pardoux", title = "Homogenization of semilinear {PDEs} with discontinuous averaged coefficients", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "18:477--18:499", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-627", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/627", abstract = "We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coefficients have averages in the Cesaro sense. In such a case, the averaged coefficients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic dierential equation (BSDE) in the Jakubowski $S$-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of ``$ L_p$-viscosity solution'' introduced in [7]. The existence of $ L_p$-viscosity solution to the averaged PDE is proved here by using BSDEs techniques.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Backward stochastic differential equations (BSDEs), $L^p$-viscosity solution for PDEs, homogenization, Jakubowski S-topology, limit in the Cesaro sense", } @Article{Denis:2009:MPC, author = "Laurent Denis and Anis Matoussi and Lucretiu Stoica", title = "Maximum Principle and Comparison Theorem for Quasi-linear Stochastic {PDE}'s", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "19:500--19:530", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-629", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/629", abstract = "We prove a comparison theorem and maximum principle for a local solution of quasi-linear parabolic stochastic PDEs, similar to the well known results in the deterministic case. The proofs are based on a version of It{\^o}'s formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary. Moreover we shortly indicate how these results generalize for Burgers type SPDEs", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic partial differential equation, It{\^o}'s formula, Maximum principle, Moser's iteration", } @Article{Toninelli:2009:CGF, author = "Fabio Toninelli", title = "Coarse graining, fractional moments and the critical slope of random copolymers", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "20:531--20:547", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-612", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/612", abstract = "For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Coarse-graining; Copolymers at Selective Interfaces; Fractional Moment Estimates", } @Article{Foondun:2009:INP, author = "Mohammud Foondun and Davar Khoshnevisan", title = "Intermittence and nonlinear parabolic stochastic partial differential equations", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "21:548--21:568", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-614", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/614", abstract = "We consider nonlinear parabolic SPDEs of the form $ \partial_t u = {\cal L} u + \sigma (u) \dot w $, where $ \dot w $ denotes space-time white noise, $ \sigma \colon R \to R $ is [globally] Lipschitz continuous, and $ \cal L $ is the $ L^2$-generator of a L'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $ \sigma $ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent, '' provided that the symmetrization of $ \cal L$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $ \cal L$ in dimension $ (1 + 1)$. When $ {\cal L} = \kappa \partial_{xx}$ for $ \kappa > 0$, these formulas agree with the earlier results of statistical physics (Kardar (1987), Krug and Spohn (1991), Lieb and Liniger (1963)), and also probability theory (Bertini and Cancrini (1995), Carmona and Molchanov (1994)) in the two exactly-solvable cases. That is when $ u_0 = \delta_0$ or $ u_0 \equiv 1$; in those cases the moments of the solution to the SPDE can be computed (Bertini and Cancrini (1995)).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Stochastic partial differential equations, Levy processes", } @Article{Gantert:2009:STR, author = "Nina Gantert and Serguei Popov and Marina Vachkovskaia", title = "Survival time of random walk in random environment among soft obstacles", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "22:569--22:593", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-631", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/631", abstract = "We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d. random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general $d$-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the ``mixed'' probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice-versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of hitting times of Branching Random Walks in Random Environment (BRWRE).", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "confinement of RWRE, survival time, quenched and annealed tails, nestling RWRE, branching random walks in random environment", } @Article{Matsui:2009:EFO, author = "Muneya Matsui and Narn-Rueih Shieh", title = "On the Exponentials of Fractional {Ornstein--Uhlenbeck} Processes", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "23:594--23:611", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-628", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/628", abstract = "We study the correlation decay and the expected maximal increment (Burkholder--Davis--Gundy type inequalities) of the exponential process determined by a fractional Ornstein--Uhlenbeck process. The method is to apply integration by parts formula on integral representations of fractional Ornstein--Uhlenbeck processes, and also to use Slepian's inequality. As an application, we attempt Kahane's T-martingale theory based on our exponential process which is shown to be of long memory.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Long memory (Long range dependence), Fractional Brownian motion, Fractional Ornstein--Uhlenbeck process, Exponential process, Burkholder--Davis--Gundy inequalities", } @Article{Chassagneux:2009:RCL, author = "Jean-Fran{\c{c}}ois Chassagneux and Bruno Bouchard", title = "Representation of continuous linear forms on the set of ladlag processes and the hedging of {American} claims under proportional costs", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "24:612--24:632", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-625", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/625", abstract = "We discuss a d-dimensional version (for l{\`a}dl{\`a}g optional processes) of a duality result by Meyer (1976) between {bounded} c{\`a}dl{\`a}g adapted processes and random measures. We show that it allows to establish, in a very natural way, a dual representation for the set of initial endowments which allow to super-hedge a given American claim in a continuous time model with proportional transaction costs. It generalizes a previous result of Bouchard and Temam (2005) who considered a discrete time setting. It also completes the very recent work of Denis, De Valli{\`e}re and Kabanov (2008) who studied c{\`a}dl{\`a}g American claims and used a completely different approach.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "American options; Randomized stopping times; transaction costs", } @Article{Kuwada:2009:CMM, author = "Kazumasa Kuwada", title = "Characterization of maximal {Markovian} couplings for diffusion processes", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "25:633--25:662", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-634", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/634", abstract = "Necessary conditions for the existence of a maximal Markovian coupling of diffusion processes are studied. A sufficient condition described as a global symmetry of the processes is revealed to be necessary for the Brownian motion on a Riemannian homogeneous space. As a result, we find many examples of a diffusion process which admits no maximal Markovian coupling. As an application, we find a Markov chain which admits no maximal Markovian coupling for specified starting points.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Maximal coupling, Markovian coupling, diffusion process, Markov chain", } @Article{Pinelis:2009:OTV, author = "Iosif Pinelis", title = "Optimal two-value zero-mean disintegration of zero-mean random variables", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "26:663--26:727", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-633", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/633", abstract = "For any continuous zero-mean random variable $X$, a reciprocating function $r$ is constructed, based only on the distribution of $X$, such that the conditional distribution of $X$ given the (at-most-)two-point set $ \{ X, r(X) \} $ is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations - of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) - go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Disintegration of measures, Wasserstein metric, Kantorovich-Rubinstein theorem, transportation of measures, optimal matching, most symmetric, hypothesis testing, confidence regions, Student's t-test, asymmetry, exact inequalities, conservative properties", } @Article{Shkolnikov:2009:CPS, author = "Mykhaylo Shkolnikov", title = "Competing Particle Systems Evolving by {I.I.D.} Increments", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "27:728--27:751", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-635", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/635", abstract = "We consider competing particle systems in $ \mathbb {R}^d $, i.e., random locally finite upper bounded configurations of points in $ \mathbb {R}^d $ evolving in discrete time steps. In each step i.i.d. increments are added to the particles independently of the initial configuration and the previous steps. Ruzmaikina and Aizenman characterized quasi-stationary measures of such an evolution, i.e., point processes for which the joint distribution of the gaps between the particles is invariant under the evolution, in case $ d = 1 $ and restricting to increments having a density and an everywhere finite moment generating function. We prove corresponding versions of their theorem in dimension $ d = 1 $ for heavy-tailed increments in the domain of attraction of a stable law and in dimension $ d \geq 1 $ for lattice type increments with an everywhere finite moment generating function. In all cases we only assume that under the initial configuration no two particles are located at the same point. In addition, we analyze the attractivity of quasi-stationary Poisson point processes in the space of all Poisson point processes with almost surely infinite, locally finite and upper bounded configurations.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Competing particle systems, Large deviations, Spin glasses", } @Article{Delyon:2009:EIS, author = "Bernard Delyon", title = "Exponential inequalities for sums of weakly dependent variables", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "28:752--28:779", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-636", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/636", abstract = "We give new exponential inequalities and Gaussian approximation results for sums of weakly dependent variables. These results lead to generalizations of Bernstein and Hoeffding inequalities, where an extra control term is added; this term contains conditional moments of the variables.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Mixing, exponential inequality; random fields; weak dependence", } @Article{Woodard:2009:SCT, author = "Dawn Woodard and Scott Schmidler and Mark Huber", title = "Sufficient Conditions for Torpid Mixing of Parallel and Simulated Tempering", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "29:780--29:804", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-638", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/638", abstract = "We obtain upper bounds on the spectral gap of Markov chains constructed by parallel and simulated tempering, and provide a set of sufficient conditions for torpid mixing of both techniques. Combined with the results of Woodard, Schmidler and Huber (2009), these results yield a two-sided bound on the spectral gap of these algorithms. We identify a persistence property of the target distribution, and show that it can lead unexpectedly to slow mixing that commonly used convergence diagnostics will fail to detect. For a multimodal distribution, the persistence is a measure of how ``spiky'', or tall and narrow, one peak is relative to the other peaks of the distribution. We show that this persistence phenomenon can be used to explain the torpid mixing of parallel and simulated tempering on the ferromagnetic mean-field Potts model shown previously. We also illustrate how it causes torpid mixing of tempering on a mixture of normal distributions with unequal covariances in $ R^M $, a previously unknown result with relevance to statistical inference problems. More generally, anytime a multimodal distribution includes both very narrow and very wide peaks of comparable probability mass, parallel and simulated tempering are shown to mix slowly.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Markov chain, rapid mixing, spectral gap, Metropolis algorithm", } @Article{Schertzer:2009:SPB, author = "Emmanuel Schertzer and Rongfeng Sun and Jan Swart", title = "Special points of the {Brownian} net", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "30:805--30:864", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-641", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/641", abstract = "The Brownian net, which has recently been introduced by Sun and Swart [16], and independently by Newman, Ravishankar and Schertzer [13], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $ \mathbb {R}^2 $ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "branching-coalescing point set.; Brownian net; Brownian web", } @Article{Caballero:2009:ABI, author = "Mar{\'\i}a Caballero and V{\'\i}ctor Rivero", title = "On the asymptotic behaviour of increasing self-similar {Markov} processes", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "31:865--31:894", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-637", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/637", abstract = "It has been proved by Bertoin and Caballero {citeBC2002} that a $ 1 / \alpha $-increasing self-similar Markov process $X$ is such that $ t^{-1 / \alpha }X(t)$ converges weakly, as $ t \to \infty, $ to a degenerate random variable whenever the subordinator associated to it via Lamperti's transformation has infinite mean. Here we prove that $ \log (X(t) / t^{1 / \alpha }) / \log (t)$ converges in law to a non-degenerate random variable if and only if the associated subordinator has Laplace exponent that varies regularly at $ 0.$ Moreover, we show that $ \liminf_{t \to \infty } \log (X(t)) / \log (t) = 1 / \alpha, $ a.s. and provide an integral test for the upper functions of $ \{ \log (X(t)), t \geq 0 \} $. Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained. In particular, these allow us to establish estimates for the left tail of some exponential functionals of subordinators. Finally, some of the implications of these results in the theory of self-similar fragmentations are discussed.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "self-similar Markov processes", } @Article{Meester:2009:USD, author = "Ronald Meester and Anne Fey-den Boer and Haiyan Liu", title = "Uniqueness of the stationary distribution and stabilizability in {Zhang}'s sandpile model", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "32:895--32:911", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-640", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/640", abstract = "We show that Zhang's sandpile model $ (N, [a, b]) $ on $N$ sites and with uniform additions on $ [a, b]$ has a unique stationary measure for all $ 0 \leq a < b \leq 1$. This generalizes earlier results of {citeanne} where this was shown in some special cases. We define the infinite volume Zhang's sandpile model in dimension $ d \geq 1$, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure $ m u$. We show that for a stationary ergodic measure $ \mu $ with density $ \rho $, for all $ \rho < \frac {1}{2}$, $ \mu $ is stabilizable; for all $ \rho \geq 1$, $ \mu $ is not stabilizable; for $ \frac {1}{2} \leq \rho < 1$, when $ \rho $ is near to $ \frac {1}{2}$ or $1$, both possibilities can occur.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Sandpile, stationary distribution, coupling, critical density, stabilizability", } @Article{Appleby:2009:SSD, author = "John Appleby and Huizhong Wu", title = "Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with applications to financial markets", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "33:912--33:959", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-642", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/642", abstract = "By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift--perturbed and exhibit asymptotic behaviour similar to standard Brownian motion. In particular sufficient conditions ensuring that these processes obey the Law of the Iterated Logarithm (LIL) are given. Ergodic--type theorems on the average growth of these non-stationary processes, which also depend on the asymptotic behaviour of the drift coefficient, are investigated. We apply these results to inefficient financial market models. The techniques extend to certain classes of finite--dimensional equation.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Brownian motion; inefficient market; Law of the Iterated Logarithm; Motoo's theorem; stationary processes; stochastic comparison principle; stochastic differential equations", } @Article{Nagahata:2009:CLT, author = "Yukio Nagahata and Nobuo Yoshida", title = "{Central Limit Theorem} for a Class of Linear Systems", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "34:960--34:977", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-644", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/644", abstract = "We consider a class of interacting particle systems with values in $ [0, \infty)^{\mathbb {Z}^d} $, of which the binary contact path process is an example. For $ d \geq 3 $ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "central limit theorem, linear systems, binary contact path process, diffusive behavior, delocalization", } @Article{Dedecker:2009:RCM, author = "J{\'e}r{\^o}me Dedecker and Florence Merlev{\`e}de and Emmanuel Rio", title = "Rates of convergence for minimal distances in the central limit theorem underprojective criteria", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "35:978--35:1011", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-648", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/648", abstract = "In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Minimal and ideal distances, rates of convergence, Martingale difference sequences", } @Article{Masson:2009:GEP, author = "Robert Masson", title = "The growth exponent for planar loop-erased random walk", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "36:1012--36:1073", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-651", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/651", abstract = "We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm--Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two dimensional discrete lattice.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "loop-erased random walk; Random walk; Schramm--Loewner evolution", } @Article{Hambly:2009:ENV, author = "Ben Hambly and Lisa Jones", title = "Erratum to {``Number Variance from a probabilistic perspective, infinite systems of independent Brownian motions and symmetric $ \alpha $-stable processes''}", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "37:1074--37:1079", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-658", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", note = "See \cite{Hambly:2007:NVP}.", URL = "http://ejp.ejpecp.org/article/view/658", abstract = "In our original paper, we provide an expression for the variance of the counting functions associated with the spatial particle configurations formed by infinite systems of independent symmetric alpha-stable processes. The formula (2.3) of the original paper, is in fact the correct expression for the expected conditional number variance. This is equal to the full variance when L is a positive integer multiple of the parameter a but, in general, the full variance has an additional bounded fluctuating term. The main results of the paper still hold for the full variance itself, although some of the proofs require modification in order to incorporate this change.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Number variance, symmetric $\alpha$-stable processes, controlled variability, Gaussian fluctuations, functional limits, long memory, Gaussian processes, fractional Brownian motion", } @Article{Schuhmacher:2009:DED, author = "Dominic Schuhmacher", title = "Distance estimates for dependent thinnings of point processes with densities", journal = j-ELECTRON-J-PROBAB, volume = "14", pages = "38:1080--38:1116", year = "2009", CODEN = "????", DOI = "https://doi.org/10.1214/EJP.v14-643", ISSN = "1083-6489", ISSN-L = "1083-6489", bibdate = "Mon Sep 1 19:06:47 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/ejp.bib", URL = "http://ejp.ejpecp.org/article/view/643", abstract = "In [Schuhmacher, Electron. J. Probab. 10 (2005), 165--201] estimates of the Barbour--Brown distance $ d_2 $ between the distribution of a thinned point process and the distribution of a Poisson process were derived by combining discretization with a result based on Stein's method. In the present article we concentrate on point processes that have a density with respect to a Poisson process, for which we can apply a corresponding result directly without the detour of discretization. This enables us to obtain better and more natural bounds in the $ d_2$-metric, and for the first time also bounds in the stronger total variation metric. We give applications for thinning by covering with an independent Boolean model and ``Matern type I'' thinning of fairly general point processes. These applications give new insight into the respective models, and either generalize or improve earlier results.", acknowledgement = ack-nhfb, ajournal = "Electron. J. Probab.", fjournal = "Electronic Journal of Probability", journal-URL = "http://ejp.ejpecp.org/", keywords = "Barbour--Brown distance; point process; point process density; Poisson process approximation; random field; Stein's method; thinning; total variation distance", } @Article{Hutzenthaler:2009:VIM, author = "Martin Hutzenthaler", title = "The {Virgin Island} Model", journal =