Exact rate of convergence of the expected $W_{2}$ distance between the empirical and true Gaussian distribution

摘要

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.