On uniform closeness of local times of Markov chains and i.i.d. sequences

TL;DR

This paper establishes uniform bounds on the total variation distance between the local times of a Markov chain and an i.i.d. sequence with the same invariant measure, using a refined soft local time method.

Contribution

It provides the first uniform (in time) bounds on the closeness of Markov chain local times to i.i.d. local times on general state spaces.

Findings

Uniform bounds on total variation distance are derived.

Results apply to Markov chains on general state spaces.

The method refines the soft local time technique.

Abstract

In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$ i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of [11].