$L^r$ -最优量化序列最大半径的渐近性
Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers
摘要：
令$P$是$\mathbb{R}^d$（配备有欧几里德范数$|\cdot|$）上的概率分布。令$ r> 0 $和$(α_n)_{n \geq1}$是$n$ -量化器的（渐近）$L^r(P)$ -最优序列.我们研究了由$ρ(α_n) = \max{|a|, a \inα_n}$为每个$n \geq1$定义的序列$(α_n)_{n \geq1}$诱导的最大半径序列的渐近行为。当$\card(\supp(P))$为无穷大时，最大半径序列随着$n$变为无穷大而变为$\sup{|x|, x \in\operatorname{supp}(P)}$。然后，我们给出了两类分布的无界支持的确切收敛速度：分布与超指数尾和分布与多项式尾。在一维的设置，一个尖锐的速率和常数提供了超指数尾部的分布。
英文摘要：
Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm $|\cdot|$). Let $ r> 0 $ and let $(α_n)_{n \geq1}$ 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 $(α_n)_{n \geq1}$ defined for every $n \geq1$ by $ρ(α_n) = \max{|a|, a \inα_n}$. 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.
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