Convergence order of the geometric mean errors for Markov-type measures

TL;DR

This paper investigates the rate at which geometric mean errors decrease for Markov-type measures on fractals, revealing that the quantization dimension of order zero is unaffected by initial probabilities under irreducibility.

Contribution

It establishes the exact convergence order of geometric mean errors for Markov measures and clarifies the influence of transition matrix irreducibility on quantization dimension.

Findings

Convergence order of geometric mean errors determined

Quantization dimension of order zero is initial-probability independent if P is irreducible

Dependence on initial probabilities when P is reducible

Abstract

We study the quantization problem with respect to the geometric mean error for Markov-type measures $\mu$ on a class of fractal sets. Assuming the irreducibility of the corresponding transition matrix $P$, we determine the exact convergence order of the geometric mean errors of $\mu$. In particular, we show that, the quantization dimension of order zero is independent of the initial probability vector when $P$ is irreducible, while this is not true if $P$ is reducible.