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Fractal functional quantization of mean-regular stochastic processes

Published online by Cambridge University Press:  22 June 2010

SIEGFRIED GRAF ,
HARALD LUSCHGY  and
GILLES PAGÈS
SIEGFRIED GRAF
Affiliation:
Universität Passau, Fakultät für Informatik und Mathematik, D-94030 Passau, Germany. e-mail: graf@fim.uni-passau.de
HARALD LUSCHGY
Affiliation:
Universität Trier, FB IV-Mathematik, D-54286 Trier, Germany. e-mail: luschgy@uni-trier.de
GILLES PAGÈS
Affiliation:
Laboratoire de Probabilités et Modèles aléatoires, UMR 7599, Université Paris 6, case 188, 4, pl. Jussieu, F-75252 Paris Cedex 5, France. e-mail: gpa@ccr.jussieu.fr
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Abstract

We investigate the functional quantization problem for stochastic processes with respect to Lp(IRd, μ)-norms, where μ is a fractal measure namely, μ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of μ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures μ we establish a (nonconstructive) link between the quantization errors of μ and the functional quantization errors of the process in the space Lp(IRd, μ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 1 , January 2011 , pp. 167 - 191
Copyright
Copyright © Cambridge Philosophical Society 2010

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