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Distortion mismatch in the quantization of probability measures

Published online by Cambridge University Press:  23 January 2008

Siegfried Graf ,
Harald Luschgy  and
Gilles Pagès
Siegfried Graf
Affiliation:
Universität Passau, Fakultät für Informatik und Mathematik, 94030 Passau, Germany; graf@fim.uni-passau.de
Harald Luschgy
Affiliation:
Universität Trier, FB IV-Mathematik, 54286 Trier, Germany; luschgy@uni-trier.de
Gilles Pagès
Affiliation:
Laboratoire de Probabilités et Modèles aléatoires, UMR 7599, Université Paris 6, case 188, 4, pl. Jussieu, 75252 Paris cedex 5, France; gpa@ccr.jussieu.fr
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Abstract

We elucidate the asymptotics of the Ls-quantization error induced by a sequence of Lr-optimal n-quantizers of a probability distribution P on $\mathbb{R}^d$ when s > r. In particular we show that under natural assumptions, the optimal rate is preserved as long as s < r+d (and for every s in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubature formulae in numerical integration on $\mathbb{R}^d$ and on the Wiener space.

Keywords

Optimal quantizationZador Theorem
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 127 - 153
Copyright
© EDP Sciences, SMAI, 2008

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