On optimal matching of Gaussian samples III

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 $\mu_n , = , \frac 1n \sum_{i=1}^n \delta_{X_i}$theassociatedempiricalmeasure,$$ $\mathbb{E}$ \big( \mathrm {W}_p^p (\mu_n , \mu )\big ) , \approx , \frac {1}{n^{p/d}} $$forany$1\leq p < d$,where$\mathrm {W}_p$isthe$p$-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.