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March 2019 Central limit theorems for empirical transportation cost in general dimension
Eustasio del Barrio, Jean-Michel Loubes
Ann. Probab. 47(2): 926-951 (March 2019). DOI: 10.1214/18-AOP1275


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.


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Eustasio del Barrio. Jean-Michel Loubes. "Central limit theorems for empirical transportation cost in general dimension." Ann. Probab. 47 (2) 926 - 951, March 2019.


Received: 1 May 2017; Revised: 1 March 2018; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053560
MathSciNet: MR3916938
Digital Object Identifier: 10.1214/18-AOP1275

Primary: 60F05 , 62E20
Secondary: 46N30

Keywords: CLT , Efron–Stein inequality , optimal matching , Optimal transportation

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 2 • March 2019
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