Limit laws of the empirical Wasserstein distance: Gaussian distributions

TL;DR

This paper establishes central limit theorems for the empirical Wasserstein distance between Gaussian samples, providing theoretical insights and applications in statistical testing and bootstrap methods.

Contribution

It introduces new CLTs for the Wasserstein distance specifically for Gaussian and elliptically symmetric distributions, based on Frechet differentiability.

Findings

Central limit theorems derived for Gaussian distributions.

Extensions to elliptically symmetric distributions discussed.

Applications include bootstrap and statistical testing methods.

Abstract

We derive central limit theorems for the Wasserstein distance between the empirical distributions of Gaussian samples. The cases are distinguished whether the underlying laws are the same or different. Results are based on the (quadratic) Frechet differentiability of the Wasserstein distance in the Gaussian case. Extensions to elliptically symmetric distributions are discussed as well as several applications such as bootstrap and statistical testing.