# On the rate of convergence in Wasserstein distance of the empirical measure

Abstract : Let $\mu_N$ be the empirical measure associated to a $N$-sample of a given probability distribution $\mu$ on $\mathbb{R}^d$. We are interested in the rate of convergence of $\mu_N$ to $\mu$, when measured in the Wasserstein distance of order $p>0$. We provide some satisfying non-asymptotic $L^p$-bounds and concentration inequalities, for any values of $p>0$ and $d\geq 1$. We extend also the non asymptotic $L^p$-bounds to stationary $\rho$-mixing sequences, Markov chains, and to some interacting particle systems.
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https://hal.archives-ouvertes.fr/hal-00915365
Contributor : Arnaud Guillin <>
Submitted on : Saturday, December 7, 2013 - 4:08:17 PM
Last modification on : Saturday, March 28, 2020 - 2:12:46 AM
Document(s) archivé(s) le : Saturday, March 8, 2014 - 12:20:19 AM

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fournier-guillin-wass.pdf
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### Identifiers

• HAL Id : hal-00915365, version 1
• ARXIV : 1312.2128

### Citation

Nicolas Fournier, Arnaud Guillin. On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, Springer Verlag, 2015, 162 (3-4), pp.707. ⟨hal-00915365⟩

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