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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60E99 None of the above but in this section
- 60G15 Gaussian processes
- 94A24 Coding theorems (Shannon theory)
- 94A34 Rate-distortion theory

**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*