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

TitleAsymptotics of the maximal radius of an ${L}^r$-optimal sequence of quantizers
Publication TypeMiscellaneous
Year of Publication2008
AuthorsGilles Pagès, and Abass Sagna
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.