Simple Transient Random Walks in One-dimensional Random Environment: the Central Limit Theorem

TL;DR

This paper establishes conditions under which a one-dimensional random walk in a random environment satisfies the Central Limit Theorem, covering i.i.d. and ergodic environments, thus advancing understanding of its probabilistic behavior.

Contribution

It provides explicit conditions on the environment ensuring the CLT for the walk's position, including for i.i.d. and ergodic cases, offering a comprehensive solution.

Findings

CLT holds under specified conditions for i.i.d. environments.

CLT also holds for uniformly ergodic environments.

Complete characterization of when the CLT applies in these settings.

Abstract

We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the Central Limit Theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.