Least upper bound of the exact formula for optimal quantization of some uniform Cantor 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 $P=\frac{1}{2}P\circ S_1^{-1}+\frac{1}{2}P\circ S_2^{-1},$ 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 $0<r<\frac{1}{2}$ and $x\in \mathbb{R}$. Then, $P$ is supported on the Cantor set generated by $S_1$ and $S_2$. The case $r=\frac{1}{3}$ 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.

arXiv commentarXiv admin overlap: text overlap with arXiv:1605.09701