论文标题
$ l^s $ rate的最佳$/$合同$ l^r $ - 最佳和贪婪的量化序列
$L^s$-rate optimality of dilated$/$contracted $L^r$-optimal and greedy quantization sequences
论文作者
论文摘要
我们研究了一些$ l^s $ rate的最佳属性的扩张/合同$ l^r $ - 最佳量化器和$ l^r $ - greedy量化序列$（α^n）_ {n \ geq 1} $的随机变量$ x $。对于$ s $的不同值，我们确定了$ l^r $ $ $ rate的最佳结果，以$（α^n_ {θ，μ}，μ}）_ {n \ geq 1} $定义的$α^n_^n_^n_ {θ，μ） α^{（n）} \} $。我们领导了一项针对径向密度分布的$ l^r $ - 最佳贪婪量化序列的特定研究，并表明它们是$ l^s $ rate在（r，r+d）$下的最佳选择。 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 $s < r+d$.我们显示，对于各种概率分布，存在一个参数$θ^*$，为此，扩张的量化序列满足所谓的{\ em $ $ l^s $ - 经验度量}定理，并将这种方法应用于数值集成。
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.
