On Optimal Matching of Gaussian Samples

Abstract

Let X1, . . .,Xn be independent random variables having as common distribution the standard Gaussian measure μ on ℝ2 and let $$ {\mu}_n=\frac{1}{n}\sum \limits_{i=1}^n{\delta}_{X_i} $$ be the associated empirical measure. We show that $$ \frac{1}{C}\frac{\log n}{n}\le $$ 𝔼 $$ \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} $$ for some numerical constant C > 0, where W2 is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan.