Quantization for infinite affine transformations

Abstract

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 $[\frac{1}{12},\frac{5}{4}]$.