Limit theorems for one-dimensional transient random walks in Markov environments*1

Abstract

We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment in the case that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer [13] for random walks in i.i.d. environments. The basic assumption is that the underlying Markov chain is irreducible and either with finite state space or with transition kernel dominated above and below by a probability measure. MSC2000: primary 60K37, 60F05; secondary 60J05, 60J80.

Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip

“…This function has in fact been introduced in [12] where it has a much simpler form; it was used in [19] and recently in [9] in a form which is very close to (2.1). The case considered in the just mentioned papers is m = 1 and hence the matrices ζ n are trivial: ζ n = 1.…”

Laws of iterated logarithm for transient random walks in random environments

“…The results have been extended to stationary and ergodic environments (cf. [1,11,16] and references therein). A general quenched central limit theorem was presented in [5].…”

Spider walk in a random environment

“…In particular, if E P (ρ 2 ) < ∞ then the central limit theorem holds with the standard normalization √ n. The limit laws of [49] are extended in [63] to environments that are stochastic functionals of either Markov processes or so called chains of infinite order. For related quenched results in the transient regime see recent articles [29,33,73] and references therein.…”