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 $X_1,\dots ,X_n$ independent random variables with common distribution the standard Gaussian measure $\mu$ on ${\mathbb{R}}^d$, $d\ge 3$, and ${\mu }_n=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{X_i}$ the associated empirical measure, $$\mathbb{E}({\mathrm{W}}_p^p({\mu }_n,\mu ))\approx \frac{1}{n^{p/d}}$$ for any $1\le p<d$, 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.