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 $\vert\cdot\vert$). Let $ r> 0 $ and let $(\alpha_n)_{n \geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\alpha_n)_{n \geq1}$ defined for every $n \geq1$ by $\rho(\alpha_n) = \max\{\vert a \vert, a \in\alpha_n \}$. When $\card(\supp(P))$ is infinite, the maximal radius sequence goes to $\sup\{ \vert x \vert, x \in\operatorname{supp}(P) \}$ as $n$ goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.

Mots clés: optimal quantization; distribution tail; function with regular variation; maximal radius of a quantizer

Domaines: Probabilités [math.PR]

Gilles Pagès, Abass Sagna. Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers. Bernoulli, 2012, 18 (1), pp.360-389. ⟨10.3150/10-BEJ333⟩. ⟨hal-00285172v2⟩