Central Limit Theorems for the Wasserstein Distance Between the Empirical and the True Distributions

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.