Abstract:

This article is a continuation of M. Ledoux papers 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 independent random variables \(X_1, \ldots, X_n\) with common distribution the standard Gaussian measure \(\mu\) on \(R^d\), \(d \geq 3\), and \(\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}\) the associated empirical measure, \[E [ \mathrm {W}_p^p (\mu_n , \mu )] \approx \frac {1}{n^{p/d}}\] for any \(1 \leq p < d\), where \(\mathrm {W}_p\) is the \(p\)th Kantorovich–Wasserstein metric. That is, in this range, the rates are the same as in the uniform case. The proof relies on the pde and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan in a compact setting.

2010 AMS Mathematics Subject Classification: Primary 60D05, 60F25; Secondary 60H15, 49J55, 58J35.

Keywords and phrases: optimal matching, empirical measure, optimal transport, Gaussian sample, Mehler kernel.