Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance

TL;DR

This paper provides simple, non-asymptotic bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance, extending existing results using concentration inequalities and applicable to various measure settings.

Contribution

It introduces straightforward deviation bounds under transport-entropy inequalities and extends to Markov chains and measures on Polish spaces with new, simplified proofs.

Findings

Deviation bounds for empirical measures in Wasserstein distance

Extension of results to Markov chain occupation measures

Applicability to Gaussian measures and diffusion processes

Abstract

We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a Markov chain are also given, under the assumption that the chain is contractive on the space of Lipschitz functions. Throughout the text, several examples are worked out, including the cases of Gaussian measures on separable Banach spaces, and laws of diffusion processes.