On optimal matching of Gaussian samples III

Abstract

This article is a continuation of the papers [8,9] in which the optimal matching problem, and the related rates of convergence of empirical measures for Gaussian samples are addressed. A further step in both the dimensional and Kantorovich parameters is achieved here, proving that, given $X_1, \ldots, X_n$ independent random variables with common distribution the standard Gaussian measure $\mu$ on $\mathbb{R}^d$, $d \geq 3$, and $\mu_n \, = \, \frac 1n \sum_{i=1}^n \delta_{X_i}$ the associated empirical measure, $$ \mathbb{E} \big( \mathrm {W}_p^p (\mu_n , \mu )\big ) \, \approx \, \frac {1}{n^{p/d}} $$ for any $1\leq p < d$, where $\mathrm {W}_p$ is the $p$-th Kantorovich metric. The proof relies on the pde and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan in a compact setting.