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 $k\geq 2$, let $\{S_j : 1\leq j\leq k\}$ 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 $P= \frac 1 k \sum_{j=1}^kP\circ S_j^{-1}$. 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 $1\leq j\leq k$. 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 $n\geq 2$. We further show that the quantization coefficient does not exist though the quantization dimension exists.