Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the mean speed of convergence of empirical and occupation measures in Wasserstein distance

Emmanuel Boissard and Thibaut Le Gouic

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 539-563.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856140

Digital Object Identifier
doi:10.1214/12-AIHP517

Mathematical Reviews number (MathSciNet)
MR3189084

Zentralblatt MATH identifier
1294.60005

Subjects
Primary: 60B10: Convergence of probability measures 65C50: Other computational problems in probability 60J05: Discrete-time Markov processes on general state spaces

Keywords
Wasserstein metrics Optimal transportation Functional quantization Transportation inequalities Markov chains Measure theory

Citation

Boissard, Emmanuel; Le Gouic, Thibaut. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 539--563. doi:10.1214/12-AIHP517. https://projecteuclid.org/euclid.aihp/1395856140


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