Constructive approach to limit theorems for recurrent diffusive random walks on a strip

TL;DR

This paper establishes constructive conditions for recurrent diffusive random walks on a strip that ensure central limit theorems, local limit theorems, and mixing properties, applicable to various environment types without stationarity assumptions.

Contribution

It introduces verifiable conditions on Green functions that guarantee key probabilistic limit theorems for a broad class of environments in recurrent diffusive random walks.

Findings

Conditions imply CLT with polynomial error bound

Conditions ensure Local Limit Theorem

Conditions lead to mixing of environment viewed by the particle

Abstract

We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.