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Asymptotic behaviour for random walks in random environments

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

S. Alili
S. Alili*
Affiliation:
Université de Cergy-Pontoise
*
∗Postal address: Département de Mathématiques, Université de Cergy-Pontoise, 2, Av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France. Email address: alili@u-cergy.fr
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Abstract

In this paper we consider limit theorems for a random walk in a random environment, (Xn). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (Xn)’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (Xn) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.

Keywords

Random walkrandom environmentbranching process

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 334 - 349
Copyright
Copyright © Applied Probability Trust 1999 

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