The Annals of Probability

Central limit theorems for empirical transportation cost in general dimension

Eustasio del Barrio and Jean-Michel Loubes

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We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on $\mathbb{R}^{d}$, with $d\geq1$. 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.

Article information

Ann. Probab., Volume 47, Number 2 (2019), 926-951.

Received: May 2017
Revised: March 2018
First available in Project Euclid: 26 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory
Secondary: 46N30: Applications in probability theory and statistics

Optimal transportation optimal matching CLT Efron–Stein inequality


del Barrio, Eustasio; Loubes, Jean-Michel. Central limit theorems for empirical transportation cost in general dimension. Ann. Probab. 47 (2019), no. 2, 926--951. doi:10.1214/18-AOP1275.

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