Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Functional quantization and small balls probabilities for Gaussian processes


Code(s) de Classification MSC:

Résumé: Quantization consists in studying the $L^r$-error induced by the approximation of a random vector $X$ by a vector (quantized version) taking a finite number $n$ of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehrinnger (2001) and Dereich et al. (2001) relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction qyantization error toward small ball probalities. This allows us to compute the exact rate of convergence to zero of the minimal $L^r$-quantization error from logarithmic small ball asymptotics and vice versa.

Mots Clés: Optimal quantizer ; asymptotic quantization error ; small ball probability ; Kolmogorov entropy ; Gaussian process

Date: 2002-06-20

Prépublication numéro: PMA-740