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.
Categories: math.PR