CENTRAL LIMIT THEOREMS FOR EMPIRICAL TRANSPORTATION COST IN GENERAL DIMENSION

We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on ℝ d , with d ≥ 1. We provide new results on the uniqueness and stability of the associated optimal transportation potentials, namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.