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Journal Articles Probability Theory and Related Fields Year : 2015

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

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Nicolas Fournier
• Function : Author
Arnaud Guillin

#### 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.

### Dates and versions

hal-00915365 , version 1 (07-12-2013)

### Identifiers

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

### Cite

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

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