A Central Limit Theorem for Wasserstein type distances between two distinct univariate distributions

In this article we study the natural nonparametric estimator of a Wasserstein type cost between two distinct continuous distributions F and G on R. The estimator is based on the order statistics of a sample having marginals F, G and any joint distribution. We prove a central limit theorem under general conditions relating the tails and the cost function. In particular, these conditions are satisfied by Wasserstein distances of order p>1and compatible classical probability distributions.