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Exact rate of convergence of the expected $$W_2$$ distance between the empirical and true Gaussian distribution. (English) Zbl 1440.62157
Summary: 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.
##### MSC:
 62G30 Order statistics; empirical distribution functions 62G20 Asymptotic properties of nonparametric inference 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles
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##### References:
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