Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Weak and strong laws of large numbers for the random normalised distortion

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