Convergence Rates for Empirical Measures of Markov Chains in Dual and Wasserstein Distances

TL;DR

This paper establishes new convergence rate bounds for empirical measures of Markov chains in dual and Wasserstein distances, matching i.i.d. rates up to logarithmic factors and including concentration inequalities.

Contribution

It introduces a novel upper bound for convergence rates of empirical measures of Markov chains using Fourier and truncation techniques, extending known i.i.d. results.

Findings

New upper bounds for expected Wasserstein distances

Convergence rates match i.i.d. rates up to logs

Concentration inequalities around the mean

Abstract

We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $\mu$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular the $1$-Wasserstein distance. The main result of this article is a new upper bound for the expected distance, which is proved by combining a Fourier expansion with a truncation argument. Our bound matches the known rates for i.i.d. random variables up to logarithmic factors. In addition, we show how concentration inequalities around the mean can be obtained.