@booklet {13,
title = {Asymptotics of the maximal radius of an ${L}^r$-optimal sequence of quantizers},
year = {2008},
abstract = {Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm). Let $r, s \superior 0$ and assume $(\alpha_n)_{n \geq 1}$ is an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. In this paper we investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\alpha_n)_{n \geq 1}$ and defined to be for every $n \geq 1$, $\ \rho(\alpha_n) = \max \{| a |, a \in \alpha_n \}$. We show that if ${\rm card(supp}(P))$ is infinite, the maximal radius sequence goes to $\sup \{| x |, x \in {\rm supp}(P) \}$ as $n$ goes to infinity. We then give the rate of convergence for two classes of distributions with unbounded support : distributions with exponential tails and distributions with polynomial tails. },
author = {Gilles Pag{\`e}s and Abass Sagna}
}