# Optimal quantization and Cantor distributions

- Authors
- Type
- Preprint
- Publication Date
- Jun 21, 2016
- Submission Date
- Jun 13, 2016
- Identifiers
- arXiv ID: 1606.04134
- Source
- arXiv
- License
- Yellow
- External links

## Abstract

The quantization scheme in probability distribution is the best approximation of the probability distribution by another probability distribution which takes only finitely many values in its support. 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, $P$ has support the Cantor set generated by $S_1$ and $S_2$. If $r=\frac 13$, then there is a closed formula for optimal quantization of the Cantor distribution. The least upper bound of $r$ for which this closed formula works was a long-time open problem. In this paper, an answer of it is given.