Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions

TL;DR

This paper investigates the optimal quantization of certain Cantor distributions generated by two similarity mappings, extending known formulas to a broader range of parameters.

Contribution

It determines the exact range of contraction parameters for which the existing quantization formula remains valid for these Cantor distributions.

Findings

Identifies the range of r-values where the Graf-Luschgy formula applies.

Extends the known exact quantization formula to a broader class of Cantor distributions.

Provides mathematical characterization of the quantization for these distributions.

Abstract

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. Let $P$ be a Borel probability measure on $\mathbb R$ such that $P=\frac 12 P\circ S_1^{-1}+\frac 12 P\circ S_2^{-1},$ where $S_1$ and $S_2$ are two contractive similarity mappings given by $S_1(x)=rx$ and $S_2(x)=rx+1-r$ for $0<r<\frac 12$ and $x\in \mathbb R$. Then, $P$ is supported on the Cantor set generated by $S_1$ and $S_2$. The case $r=\frac 13$ was treated by Graf and Luschgy who gave an exact formula for the unique optimal quantization of the Cantor distribution $P$ (Math. Nachr., 183 (1997), 113-133). In this paper, we compute the precise range of $r$-values to which Graf-Luschgy formula extends.