Quantization for infinite affine transformations

TL;DR

This paper studies the quantization of probability distributions generated by infinite affine transformations, providing explicit calculations of optimal means, quantization errors, and analyzing the quantization dimension and coefficients.

Contribution

It introduces a method to compute optimal quantization for distributions from infinite affine systems and characterizes their quantization dimension and coefficients.

Findings

Optimal n-means and quantization errors are explicitly calculated.

The distribution matches that of a direct product of Cantor distributions.

Quantization dimension exists and is finite, but the quantization coefficient does not exist, with bounds in [1/12, 5/4].

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