Exact convergence order of the $L_r$-quantization error for Markov-type measures

TL;DR

This paper determines the precise rate at which the $L_r$-quantization error converges for Markov-type measures on graph-directed sets, especially in cases with infinite quantization coefficients.

Contribution

It establishes the exact convergence order of the $L_r$-quantization error for Markov-type measures under separation conditions, advancing understanding of quantization asymptotics.

Findings

Exact convergence order of the $L_r$-quantization error is derived.

Provides asymptotic behavior of quantization error when the quantization coefficient is infinite.

Results apply to graph-directed sets with Markov measures.

Abstract

Let $E$ be a graph-directed set associated with a di-graph $G$. Let $\mu$ be a Markov-type measure on $E$. Assuming a separation condition for $E$, we determine the exact convergence order of the $L_r$-quantization error for $\mu$. This result provides us with accurate information on the asymptotics of the quantization error, especially when the quantization coefficient is infinite.