The quantization of the standard triadic Cantor distribution

TL;DR

This paper studies the optimal quantization of the standard triadic Cantor distribution, analyzing the best finite approximations and quantization errors, and reveals that the quantization coefficient does not exist despite the quantization dimension existing.

Contribution

It provides a detailed analysis of the optimal n-means and quantization errors specifically for the triadic Cantor distribution, a case not previously fully explored.

Findings

Optimal sets of n-means are characterized for the triadic Cantor distribution.

Quantization errors are explicitly computed for all n ≥ 2.

Quantization coefficient does not exist, but quantization dimension does.

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.