A Central Limit Theorem for Semidiscrete Wasserstein Distances

Abstract

We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{n}\{{\mathcal{T}}_c(P_n,Q)-{\mathcal{W}}_c(P,Q)\}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported. We show that the asymptotic distribution is the supremun of a centered Gaussian process which is Gaussian under some additional conditions on the probability $Q$ and on the cost. Such results imply the central limit theorem for the $p$-Wassertein distance, for $p\ge 1$. Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials.