The quantization of the standard triadic Cantor distribution

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. For a given $k\ge 2$, let $\{S_j:1\le j\le k\}$ be a set of $k$ contractive similarity mappings such that $S_j(x)=\frac{1}{2k-1}x+\frac{2(j-1)}{2k-1}$ for all $x\in \mathbb{R}$, and let $P=\frac{1}{k}{\sum }_{j=1}^{k}P\circ S_j^{-1}$. Then, $P$ is a unique Borel probability measure on $\mathbb{R}$ such that $P$ has support the Cantor set generated by the similarity mappings $S_j$ for $1\le j\le k$. In this paper, for the probability measure $P$, when $k=3$, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all $n\ge 2$. We further show that the quantization coefficient does not exist though the quantization dimension exists.