Behavior of the empirical Wasserstein distance in R^d under moment conditions

TL;DR

This paper investigates the behavior of the Wasserstein distance between empirical and true distributions in R^d, providing deviation inequalities, moment bounds, and almost sure results under moment conditions.

Contribution

It introduces new deviation inequalities and bounds for the Wasserstein distance under minimal moment assumptions, enhancing understanding of its probabilistic behavior.

Findings

Derived deviation inequalities for Wasserstein distance

Established moment bounds under moment conditions

Proved almost sure convergence results

Abstract

We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order p $\in$ [1, $\infty$) between the empirical measure of independent and identically distributed R d-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order rp for some r > 1, and we discuss the optimality of the bounds. Mathematics subject classification. 60B10, 60F10, 60F15, 60E15.