Convergence of the empirical measure in expected Wasserstein distance: non asymptotic explicit bounds in $\mathbb{R}^d$

TL;DR

This paper establishes explicit non-asymptotic bounds on how quickly the empirical measure converges to the true distribution in expected Wasserstein distance in multi-dimensional space.

Contribution

It provides the first explicit non-asymptotic bounds with constants for the convergence rate of empirical measures in Wasserstein distance in .

Findings

Explicit bounds depend on sample size and dimension

Bounds are non-asymptotic with clear constants

Results applicable to i.i.d. samples in 

Abstract

We provide some non asymptotic bounds, with explicit constants, that measure the rate of convergence, in expected Wasserstein distance, of the empirical measure associated to an i.i.d. $N$-sample of a given probability distribution on $\mathbb{R}^d$.