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A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  30 January 2018

Hoang-Chuong Lam
Hoang-Chuong Lam*
Affiliation:
Can Tho University and Ben Gurion University
*
∗ Postal address: Department of Mathematics, Can Tho University, Can Tho City, Vietnam. Email address: lhchuong@ctu.edu.vn
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Abstract

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The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (Pω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

Keywords

Quenched central limit theoremreversible random walk in random environment

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J60: Diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1051 - 1064
Copyright
© Applied Probability Trust 

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