ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: PS, 2008, Vol. 12, p. 127-153
DOI: 10.1051/ps:2007044

Distortion mismatch in the quantization of probability measures

Siegfried Graf¹, Harald Luschgy² and Gilles Pagès³

¹ Universität Passau, Fakultät für Informatik und Mathematik, 94030 Passau, Germany; graf@fim.uni-passau.de
² Universität Trier, FB IV-Mathematik, 54286 Trier, Germany; luschgy@uni-trier.de
³ 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

Received June 2, 2006. Revised November 24, 2006. Published online 23 January 2008

Abstract
We elucidate the asymptotics of the L^s-quantization error induced by a sequence of L^r-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.

Mathematics Subject Classification. 60G15, 60G35, 41A25

Key words: Optimal quantization, Zador Theorem

© EDP Sciences, SMAI 2008