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April 1999 Central Limit Theorems for the Wasserstein Distance Between the Empirical and the True Distributions
Eustasio del Barrio, Evarist Giné, Carlos Matrán
Ann. Probab. 27(2): 1009-1071 (April 1999). DOI: 10.1214/aop/1022677394

Abstract

If $X$ is integrable, $F$ is its cdf and $F_n$ is the empirical cdf based on an i.i.d. sample from $F$, then the Wasserstein distance between $F_n$ and $F$, which coincides with the $L_1$ norm $\int_-\infty^\infty|F_n(t)|dt$ of the centered empirical process, tends to zero a.s. The object of this article is to obtain rates of convergence and distributional limit theorems for this law of large numbers or, equivalently, stochastic boundedness and distributional limit theorems for the $L_1$ norm of the empirical process. Some limit theorems for the Ornstein–Uhlenbeck process are also derived as a by-product.

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Eustasio del Barrio. Evarist Giné. Carlos Matrán. "Central Limit Theorems for the Wasserstein Distance Between the Empirical and the True Distributions." Ann. Probab. 27 (2) 1009 - 1071, April 1999. https://doi.org/10.1214/aop/1022677394

Information

Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0958.60012
MathSciNet: MR1698999
Digital Object Identifier: 10.1214/aop/1022677394

Subjects:
Primary: 60F17, 62E20
Secondary: 60B12, 60J65

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 2 • April 1999
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