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.
Publié le : 2015-08-05
Classification:
Empirical measure,
Sequence of i.i.d. random variables,
Wasserstein distance,
Concentration inequalities,
Quantization,
Markov chains,
60F25, 60F10, 65C05, 60E15, 65D32,
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR],
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST],
[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
@article{hal-00915365,
author = {Fournier, Nicolas and Guillin, Arnaud},
title = {On the rate of convergence in Wasserstein distance of the empirical measure},
journal = {HAL},
volume = {2015},
number = {0},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00915365}
}
Fournier, Nicolas; Guillin, Arnaud. On the rate of convergence in Wasserstein distance of the empirical measure. HAL, Tome 2015 (2015) no. 0, . http://gdmltest.u-ga.fr/item/hal-00915365/