On the mean speed of convergence of empirical and occupation measures in Wasserstein distance

TL;DR

This paper establishes non-asymptotic bounds on how quickly empirical and occupation measures converge in Wasserstein distance, with implications for quantization and approximation of probability measures, including in infinite-dimensional spaces.

Contribution

It provides new non-asymptotic convergence bounds for empirical and occupation measures in Wasserstein distance, connecting these rates to optimal quantization rates in various settings.

Findings

Convergence rates match known optimal quantization rates.

Results apply to infinite-dimensional Gaussian measures.

Rates are close for empirical and occupation measures.

Abstract

In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.