Electronic Journal of Probability

Behavior of the empirical Wasserstein distance in ${\mathbb R}^d$ under moment conditions

Jérôme Dedecker and Florence Merlevède

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We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order $p\in [1, \infty )$ between the empirical measure of independent and identically distributed ${\mathbb R}^d$-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order $rp$ for some $r>1$, and we discuss the optimality of the bounds.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 6, 32 pp.

Received: 24 July 2018
Accepted: 14 January 2019
First available in Project Euclid: 14 February 2019

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Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures 60F10: Large deviations 60F15: Strong theorems 60E15: Inequalities; stochastic orderings

empirical measure Wasserstein distance independent and identically distributed random variables deviation inequalities moment inequalities almost sure rates of convergence

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Dedecker, Jérôme; Merlevède, Florence. Behavior of the empirical Wasserstein distance in ${\mathbb R}^d$ under moment conditions. Electron. J. Probab. 24 (2019), paper no. 6, 32 pp. doi:10.1214/19-EJP266. https://projecteuclid.org/euclid.ejp/1550113245

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