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

Emmanuel Boissard; Thibaut Le Gouic

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 539-563
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Boissard, Emmanuel, and 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 50.2 (2014): 539-563. <http://eudml.org/doc/271944>.

@article{Boissard2014,
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.},
author = {Boissard, Emmanuel, Le Gouic, Thibaut},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Wasserstein metrics; optimal transportation; functional quantization; transportation inequalities; Markov chains; measure theory},
language = {eng},
number = {2},
pages = {539-563},
publisher = {Gauthier-Villars},
title = {On the mean speed of convergence of empirical and occupation measures in Wasserstein distance},
url = {http://eudml.org/doc/271944},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Boissard, Emmanuel
AU - Le Gouic, Thibaut
TI - On the mean speed of convergence of empirical and occupation measures in Wasserstein distance
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 539
EP - 563
AB - 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.
LA - eng
KW - Wasserstein metrics; optimal transportation; functional quantization; transportation inequalities; Markov chains; measure theory
UR - http://eudml.org/doc/271944
ER -

References

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