# 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
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.