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Uniform decomposition of probability measures: quantization, clustering and rate of convergence

Part of: Limit theorems Distribution theory - Probability Probability theory on algebraic and topological structures Distribution theory

Published online by Cambridge University Press:  16 January 2019

Julien Chevallier
Julien Chevallier*
Affiliation:
Université Grenoble Alpes
*
* Postal address: Université Grenoble Alpes, CNRS, LJK, 38000 Grenoble, France. Email address: julien.chevallier1@univ-grenoble-alpes.fr
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Abstract

The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimension d=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimension d≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.

Keywords

Uniform approximationWasserstein distancerate of convergencequantizationclustering

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E17: Approximations to distributions (nonasymptotic) 60B10: Convergence of probability measures 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1037 - 1045
Copyright
Copyright © Applied Probability Trust 2018 

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