Central limit theorems for biased randomly trapped random walks on Z

TL;DR

This paper establishes central limit theorems for biased randomly trapped random walks on the integer lattice, analyzing both annealed and quenched cases, with applications to walks on Galton-Watson trees.

Contribution

It introduces CLTs for biased trapped walks in one dimension, including environment-dependent centering in quenched cases, and applies results to Galton-Watson trees.

Findings

Proved annealed CLT for finite second moment trapping times.

Derived environment-dependent centering for quenched CLT.

Bounded the bias needed for CLT on Galton-Watson trees.

Abstract

We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have finite second moment. In a quenched environment, an environment dependent centring is determined which is necessary to achieve a central limit theorem. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive and prove a tight bound on the bias required to obtain such limiting behaviour.