Exact rate of convergence of the mean Wasserstein distance between the empirical and true Gaussian distribution

We study the Wasserstein distance \(W_2\) for Gaussian samples. We establish the exact rate of convergence \(\sqrt{\log\log n/n}\) of the expected value of the \(W_2\) 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 \(1/\sqrt{n}\) in the case of two correlated Gaussian samples.