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February 2020 Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces
Jing Lei
Bernoulli 26(1): 767-798 (February 2020). DOI: 10.3150/19-BEJ1151

Abstract

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

Citation

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Jing Lei. "Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces." Bernoulli 26 (1) 767 - 798, February 2020. https://doi.org/10.3150/19-BEJ1151

Information

Received: 1 January 2019; Revised: 1 July 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140516
MathSciNet: MR4036051
Digital Object Identifier: 10.3150/19-BEJ1151

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 1 • February 2020
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