Exact rate of convergence of the expected W2 distance between the empirical and true gaussian distribution

Abstract

We study the Wasserstein distance W2 for Gaussian samples. We establish the exact rate of convergence √log log n/n of the expected value of the W2 distance between the empirical and true c.d.f.’s for the normal distribution. We also show that the rate of weak convergence is unexpectedly 1/√n in the case of two correlated Gaussian samples.