Two-boundary problems for a Poisson process with exponentially distributed component

For a Poisson process with exponentially distributed negative component, we obtain integral transforms of the joint distribution of the time of the first exit from an interval and the value of the jump over the boundary at exit time and the joint distribution of the time of the first hit of the interval and the value of the process at this time. On the exponentially distributed time interval, we obtain distributions of the total sojourn time of the process in the interval, the joint distribution of the supremum, infimum, and value of the process, the joint distribution of the number of upward and downward crossings of the interval, and generators of the joint distribution of the number of hits of the interval and the number of jumps over the interval. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 922–953, July, 2006.     

一类随机环境下随机游动的常返性

就一类平稳环境θ下随机流动{Xn}n∈z 建立相应的Markov-双链{ηn}n∈z ={(xn,Tnθ)}n∈z ,并给出在该平稳环境θ下{xn}n∈z 为常返链的条件.     

Moderate deviations for a random walk in random scenery

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér’s condition. We prove moderate deviation principles in dimensions d≥2$d\ge 2$, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d≥4$d\ge 4$ we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d≥3$d\ge 3$, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst for d=2$d=2$ we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.     

Limit theorem for a random walk in a reversible random environment

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 4, pp. 627–644, October–December, 1989.     

A monotonicity property for random walk in a partially random environment

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in ${\mathbb{Z}}^d$ that is an extension of a result of Bolthausen et al. (2003) [4]. We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the first coordinate of the speed of the random walk under some strong assumptions on the distribution of the environment.     

Shift-coupling and a zero-one law for random walk in random environment

This paper discusses several aspects of shift-coupling for random walk in random environment.     

Diffusivity of a random walk on random walks

We consider a random walk with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor with respect to the case of the classical simple random walk without constraint. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 267–283, 2015     

Slowdown and neutral pockets for a random walk in random environment

We consider a d-dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift “pointing to the right”. We obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion, both under the annealed and quenched measures. These estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk. Received: 12 March 1998 / Revised version: 19 February 1999     

Hitting times with taboo for a random walk

For a symmetric homogeneous and irreducible random walk on the d-dimensional integer lattice, which have a finite variance of jumps, we study passage times (taking values in [0,??]) determined by a starting point x, a hitting state y, and a taboo state z. We find the probability that these passage times are finite, and study the distribution tail. In particular, it turns out that, for the above-mentioned random walks on ? ^d except for a simple random walk on ?, the order of the distribution tail decrease is specified by dimension d only. In contrast, for a simple random walk on ?, the asymptotic properties of hitting times with taboo essentially depend on mutual location of the points x, y, and z. These problems originated in recent study of a branching random walk on ? ^d with a single source of branching.     

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed.     

Product-form characterization for a two-dimensional reflecting random walk

We consider a two-dimensional skip-free reflecting random walk on the non-negative integers, which is referred to as a 2-d reflecting random walk. We give necessary and sufficient conditions for the stationary distribution to have a product-form. We also derive simpler sufficient conditions for the product-form for a restricted class of 2-d reflecting random walks. We apply these results and obtain a product-form approximation of the stationary distribution through a suitable modification of the parameters of the random walk.     

Eigenvectors for a random walk on a hyperplane arrangement

We find explicit eigenvectors for the transition matrix of the Bidigare–Hanlon–Rockmore random walk, from Bidigare et al. (1999) [1]. This is accomplished by using Brown and Diaconis? (1998) analysis in [3] of the stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gel?fand and Varchenko (1987) [10].     

Revisits for transient random walk

For a random walk on the integers define R_n as the number of (distinct) states visited in the first n steps and Z_n as the number of states visited in the first n steps which are never revisited. Here we deal with transient walks. The increments of Z_n form a stationary process and various central limit results and an iterated logarithm result are obtained for Z_n from known results on stationary processes. Furthermore, the limit behaviour of R_n is closely related to that of Z_n; this relationship is elucidated and corresponding limit results for R_n are then read off from those for Z_n.     

Return probabilities for random walk on a half-line

A random walk with reflecting zone on the nonnegative integers is a Markov chain whose transition probabilitiesq(x, y) are those of a random walk (i.e.,q(x, y)=p(y–x)) outside a finite set {0, 1, 2,...,K}, and such that the distributionq(x,·) stochastically dominatesp(·–x) for everyx{0, 1, 2,..., K}. Under mild hypotheses, it is proved that when xp _x>0, the transition probabilities satisfyq ⁿ(x, y)C_xyR^–nn^–3/2 asn, and when xp _x=0,q ⁿ(x, y)C_xyn^–1/2.Supported by National Science Foundation Grant DMS-9307855.     

Eigenvectors for a random walk on a left-regular band

We present a simple construction of the eigenvectors for the transition matrices of random walks on a class of semigroups called left-regular bands. These walks were introduced and analyzed by Brown, and they include the hyperplane chamber walks of Bidigare, Hanlon and Rockmore. This construction leads to new concise proofs of several of the known results about these walks. We also explain how tools from poset topology can be used to extract an eigenbasis for the transition matrices of the hyperplane chamber walks, and indicate the connection with a method recently described by Denham.