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.
@article{1022677394,
author = {del Barrio, Eustasio and Gin\'e, Evarist and Matr\'an, Carlos},
title = {Central Limit Theorems for the Wasserstein Distance Between the
Empirical and the True Distributions},
journal = {Ann. Probab.},
volume = {27},
number = {1},
year = {1999},
pages = { 1009-1071},
language = {en},
url = {http://dml.mathdoc.fr/item/1022677394}
}
del Barrio, Eustasio; Giné, Evarist; Matrán, Carlos. Central Limit Theorems for the Wasserstein Distance Between the
Empirical and the True Distributions. Ann. Probab., Tome 27 (1999) no. 1, pp. 1009-1071. http://gdmltest.u-ga.fr/item/1022677394/