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:**

- 60F25 $L^p$-limit theorems
- 60F15 Strong theorems
- 94A29 Source coding

**Résumé:** We present some convergence results about the
distortion $\ds{D_{\mu,N,r}^{\nu}}$
related to the Vorono\"{\i} vector quantization of a
$\mu$-distributed random variable using $N$ i.i.d. $\nu$-distributed
codes. A weak law of large numbers for $\ds{N^{\frac{r}{d}}
D_{\mu,N,r}^{\nu}}$ is derived essentially
under a $\mu$-integrability condition
on a negative power of a $\delta$-lower Radon-Nicodym derivative of
$\nu$. Assuming in addition that the probability measure $\mu$
has a bounded $\varepsilon$-potential, we obtain a strong
law of large numbers for $\ds{N^{\frac{r}{d}}
D_{\mu,N,r}^{\nu}}$.
In particular, we show that the random distortion and the optimal
distortion vanish almost surely at the same rate.

**Mots Clés:** *quantization ; distortion ; law of large numbers*

**Date:** 2000-03-30

**Prépublication numéro:** *PMA-582*

**Revised version :**PMA-582.ps