Central Limit Theorem for empirical transportation cost in general dimension

Eustasio del Barrio (1) , Jean-Michel Loubes (2)

1 UVa - Universidad de Valladolid [Valladolid]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: optimal transportation; CLT; optimal matching; Efron-Stein inequality

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Eustasio del Barrio, Jean-Michel Loubes. Central Limit Theorem for empirical transportation cost in general dimension. Annals of Probability, 2019, 2. ⟨hal-01517192v3⟩