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

### Quantization of probability distributions under norm-based distortion measures

Auteur(s):

Code(s) de Classification MSC:

• 60E99 None of the above, but in this section
Résumé: For a probability measure $P$ on $\R^d$ and $n\!\! \in \! \N$ consider $e_n = \inf \displaystyle \int \min_{a \in \alpha} V(\| x-a \| )dP(x)$ where the infimum is taken over all subsets $\alpha$ of $\R^d$ with $\mbox{card} (\alpha) \leq n$ and $V$ is a nondecreasing function. Under certain conditions on $V$, we derive the precise $n$-asymptotics of $e_n$ for nonsingular and for (singular) self-similar distributions $P$ and we find the asymptotic performance of optimal quantizers using weighted empirical measures.