Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers

Gilles Pagès (1) , Abass Sagna (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm). Let $ r,s > 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 $n \geq 1$, $\ \rho(\alpha_n) = \max \{\vert a \vert, \ a \in \alpha_n \}$. We show that if ${\rm card(supp}(P))$ is infinite, the maximal radius sequence goes to $\sup \{ \vert x \vert, x \in {\rm supp}(P) \}$ 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.

Mots clés: random quantization; optimal quantization; maximal radius; $(r; s)$-distribution; random quantization.

Domaines: Probabilités [math.PR]

Gilles Pagès, Abass Sagna. Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers. 2008. ⟨hal-00285172v1⟩