On the mean speed of convergence of empirical and occupation measures in Wasserstein distance
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, p. 539-563

In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.

Dans ce travail, on exhibe des bornes non asymptotiques pour la vitesse de convergence en moyenne de la mesure empirique dans la loi des grands nombres, en distance de Wasserstein. On considère également la mesure d'occupation d'une chaîne de Markov ergodique. L'une des motivations est l'approximation d'une mesure de probabilité par des mesures à support fini (le problème de la quantification). On détermine que les taux de convergence des mesures empiriques ou des mesures d'occupation correspondent dans plusieurs cas aux taux de quantification optimale déjà établis par ailleurs. Ce fait est notamment établi pour des mesures gaussiennes dans des espaces de dimension infinie.

DOI : https://doi.org/10.1214/12-AIHP517
Classification:  60B10,  65C50,  60J05
Keywords: Wasserstein metrics, optimal transportation, functional quantization, transportation inequalities, Markov chains, measure theory
@article{AIHPB_2014__50_2_539_0,
     author = {Boissard, Emmanuel and Le Gouic, Thibaut},
     title = {On the mean speed of convergence of empirical and occupation measures in Wasserstein distance},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     pages = {539-563},
     doi = {10.1214/12-AIHP517},
     zbl = {1294.60005},
     mrnumber = {3189084},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_2_539_0}
}
Boissard, Emmanuel; Le Gouic, Thibaut. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 539-563. doi : 10.1214/12-AIHP517. http://www.numdam.org/item/AIHPB_2014__50_2_539_0/

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