On Optimal Matching of Gaussian Samples

Abstract

Let X₁, . . .,X_n 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

\( \frac{1}{C}\frac{\log n}{n}\le \) ???? \( \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} \)

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.