Quantization for a probability distribution generated by an infinite iterated function system

TL;DR

This paper develops an induction formula to determine optimal quantization points and errors for probability measures generated by infinite iterated function systems, advancing understanding of their approximation properties.

Contribution

It introduces a novel induction formula for optimal quantization of measures from infinite iterated function systems, providing a systematic way to find optimal sets and errors.

Findings

Derived an induction formula for optimal n-means and quantization error.

Provided results and observations on optimal sets for all n ≥ 2.

Enhanced understanding of quantization for complex probability measures.

Abstract

Quantization for probability distributions concerns the best approximation of a $d$-dimensional probability distribution $P$ by a discrete probability with a given number $n$ of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on $\mathbb R$. For such a probability measure $P$, an induction formula to determine the optimal sets of $n$-means and the $n$th quantization error for every natural number $n$ is given. In addition, using the induction formula we give some results and observations about the optimal sets of $n$-means for all $n\geq 2$.