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

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