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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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