zbMATH — the first resource for mathematics

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.
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
Full Text: DOI Euclid
[1] P. Berthet and J-C. Fort. Weak convergence of Wasserstein type distances. hal-01838700v2, 2018 and arXiv:1911.02389, 2019.
[2] P. Berthet, J-C. Fort, and T. Klein. A central limit theorem for Wasserstein type distances between two different real distributions. To appear in Ann. Inst. Henri Poincaré Probab. Stat., hal-01526879, 2018.
[3] P. J. Bickel and W. R. van Zwet. Asymptotic expansions for the power of distribution free tests in the two-sample problem. Ann. Statist., 6(5):937-1004, 1978. · Zbl 0378.62047
[4] S. G. Bobkov and M. Ledoux. One-dimensional empirical measures, order statistics and Kantorovich transport distances. To appear in: Memoirs of the AMS, Preprint 2016. · Zbl 1454.60007
[5] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. Math., 30(2):207-216, 1975. · Zbl 0292.60004
[6] H. Cramer. Mathematical methods of statistics. Princeton Mathematical Series; 9. 1945. · Zbl 0985.62001
[7] M. Csörgö and L. Horváth. Weighted approximations in probability and statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. 1993. · Zbl 0770.60038
[8] M. Csörgö, L. Horváth, and Q.-M. Shao. Convergence of integrals of uniform empirical and quantile processes. Stochastic Processes and their Applications, 45(2):283-294, 1993. · Zbl 0784.60038
[9] E. del Barrio, E. Giné, and F. Utzet. Asymptotics for \(L_2\) functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances. Bernoulli, 11(1):131-189, 2005. · Zbl 1063.62072
[10] M. Ledoux and M. Talagrand. Probability in Banach spaces. Classics in Mathematics. Springer-Verlag, Berlin, 2011. · Zbl 1226.60003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.