Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension

Jérôme Dedecker; Aurélie Fischer; Bertrand Michel

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arXiv · 2601.11133

Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension

Jérôme Dedecker·Aurélie Fischer·Bertrand Michel

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Abstract

We give concentration inequalities in Wasserstein distance for the empirical measure of a sequence of independent and identically distributed random variables with values in a Polish space E. These inequalities involve the covering dimension of the support of the distribution of the variables. More precisely, we obtain a complete extension of the concentration inequalities of Fournier and Guillin [2015] in the case where E = R^d , in which the covering dimension replaces the dimension of the ambient space E.

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Jérôme Dedecker, Aurélie Fischer, Bertrand Michel. 2026-01-16. Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension. https://arxiv.org/abs/2601.11133

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