Coupling for features of random walks

TL;DR

This paper explores how coupling techniques can be used to bound the mixing times of statistics derived from Markov chains, extending traditional methods focused on the chains themselves.

Contribution

It introduces a novel approach to analyze the mixing of statistics on Markov chains using coupling, providing new bounds beyond standard chain mixing times.

Findings

Coupling can effectively bound the mixing time of Markov chain statistics.

The method offers potentially tighter bounds for specific statistics.

The approach broadens the applicability of coupling in Markov chain analysis.

Abstract

We use coupling to study the time taken until the distribution of a statistic on a Markov chain is close to its stationary distribution. Coupling is a common technique used to obtain upper bounds on mixing times of Markov chains, and we explore how this technique may be used to obtain bounds on the mixing of a statistic instead.