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 $\sqrt{\log \log n\mathrm{/}n}$ of the expected value of the $W_2$ distance between the empirical and true $c\mathrm{.}d\mathrm{.}f\mathrm{.}$'s for the normal distribution. We also show that the rate of weak convergence is unexpectedly $1\mathrm{/}\sqrt{n}$ in the case of two correlated Gaussian samples.