If X is integrable, F is its cdf and Fₙ is the empirical cdf based on an i.i.d. sample from F, then the Wasserstein distance between Fₙ and F, which coincides with the L₁ norm ∫₋∞^∞|Fₙ(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 L₁ norm of the empirical process. Some limit theorems for the Ornstein–Uhlenbeck process are also derived as a by-product.
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Barrio et al. (1999) studied this question.