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 {Sij} on R2 with associated probabilities {pij} such that pij>0 for all i,j∈N and ∑i,j=1∞pij=1. For such a probability measure P, the optimal sets of n-means and the nth 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 [112,54].