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

TL;DR

This paper studies the asymptotic behavior of the maximal radius of $L^r$-optimal quantizers for distributions on $R^d$, providing exact convergence rates for distributions with hyper-exponential and polynomial tails.

Contribution

It characterizes the asymptotic rate of the maximal radius for $L^r$-optimal quantizers, including sharp rates and constants for hyper-exponential tail distributions in one dimension.

Findings

Maximal radius converges to the support supremum for infinite support distributions.

Exact convergence rates are derived for hyper-exponential tail distributions.

Convergence rate and constant are sharp in the one-dimensional hyper-exponential case.

Abstract

Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm $|\cdot|$). 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 $\rho(\alpha_n) = \max{|a|, a \in\alpha_n}$. When $\card(\supp(P))$ is infinite, the maximal radius sequence goes to $\sup{|x|, x \in\operatorname{supp}(P)}$ 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.