无限仿射变换的量子化
Quantization for infinite affine transformations
摘要：
概率分布的量化是指通过有限集支持的离散概率来估计给定概率的想法。在这篇文章中，我们考虑一个概率分布所产生的无限系统的仿射变换$\{S_{ij}\}$的$\mathbb R^2$与相关的概率$\{p_{ij}\}$，使$p_{ij}>0$的所有$i, j\in \mathbb N$和$\sum_{i, j=1}^\infty p_{ij}=1$。对于这样的概率测量$P$，针对每个自然数$n$计算$n$ -均值和PDMATHABMT_th量化误差的最优集合。它表明，这样的概率测度的分布是相同的康托分布的直积。此外，还证明了量化维数$D(P)$存在且有限，而$D(P)$维量化系数不存在，且$D(P)$维上下量化系数位于闭区间$[\frac{1}{12}, \frac{5}{4}]$内.
英文摘要：
Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite set. In this article, we consider a probability distribution generated by an infinite system of affine transformations $\{S_{ij}\}$ on $\mathbb R^2$ with associated probabilities $\{p_{ij}\}$ such that $p_{ij}>0$ for all $i, j\in \mathbb N$ and $\sum_{i, j=1}^\infty p_{ij}=1$. For such a probability measure $P$, the optimal sets of $n$-means and the $n$th quantization error are calculated for every natural number $n$. It is shown that the distribution of such a probability measure is the same as that of the direct product of the Cantor distribution. In addition, it is proved that the quantization dimension $D(P)$ exists and is finite; whereas, the $D(P)$-dimensional quantization coefficient does not exist, and the $D(P)$-dimensional lower and the upper quantization coefficients lie in the closed interval $[\frac{1}{12}, \frac{5}{4}]$.
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