扩张PDMATHAAB收缩$L^r$ -最优贪婪量化序列的$L^s$ -速率最优性
$L^s$-rate optimality of dilated$/$contracted $L^r$-optimal and greedy quantization sequences
摘要：
研究了随机变量$X$的扩张/收缩$L^r$ -最优量化器和$L^r$ -贪婪量化序列$(α^n)_{n \geq 1}$的$L^s$ -速率最优性.对于不同的$s$值，我们建立了由$α^n_{θ,μ}=\{μ+θ(α_i-μ), α_i \in α^{(n)}\}$定义的$L^r$-最优膨胀/收缩贪婪量化序列$(α^n_{θ,μ})_{n \geq 1}$的$L^s$ -速率最优性结果。我们领导一个具体的研究$L^r$ -最佳贪婪量化序列的径向密度分布，并表明，他们是$L^s$率最佳的$s \in (r,r+d)$下的一些时刻的假设。基于建立在$\cite{Sagna08}$的$L^r$ -最佳量化器的结果，我们表明，对于一个较大的类的分布，膨胀$(α^n_{θ,μ})_{n \geq 1}$的$L^r$ -最佳量化器是$L^s$ -率最佳的$s < r+d$。我们表明，对于各种概率分布，存在一个参数$θ^*$的膨胀的量化序列满足所谓的{ \em $L^s$ -经验措施}定理，并提出了应用这种方法的数值积分。
英文摘要：
We investigate some $L^s$-rate optimality properties of dilated/contracted $L^r$-optimal quantizers and $L^r$-greedy quantization sequences $(α^n)_{n \geq 1}$ of a random variable $X$. We establish, for different values of $s$, $L^s$-rate optimality results for $L^r$-optimally dilated/contracted greedy quantization sequences $(α^n_{θ,μ})_{n \geq 1}$ defined by $α^n_{θ,μ}=\{μ+θ(α_i-μ), α_i \in α^{(n)}\}$. We lead a specific study for $L^r$-optimal greedy quantization sequences of radial density distributions and show that they are $L^s$-rate optimal for $s \in (r,r+d)$ under some moment assumption. Based on the results established in $\cite{Sagna08}$ for $L^r$-optimal quantizers, we show, for a larger class of distributions, that the dilatation $(α^n_{θ,μ})_{n \geq 1}$ of an $L^r$-optimal quantizer is $L^s$-rate optimal for $s < r+d$. We show, for various probability distributions, that there exists a parameter $θ^*$ for which the dilated quantization sequence satisfy the so-called {\em $L^s$-empirical measure} theorem and present an application of this approach to numerical integration.
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