On some mixing properties of copula-based Markov chains

TL;DR

This paper investigates the mixing properties of copula-based Markov chains, providing new tools for analysis, exploring the effects of perturbations, and correcting previous misconceptions about $\psi$-mixing.

Contribution

It introduces new methods to verify $\psi$-mixing properties, analyzes the impact of perturbations on these chains, and corrects prior inaccuracies in the literature.

Findings

Perturbations of $\psi'$-mixing chains remain $\psi'$-mixing.

Perturbations of $\psi$-mixing chains do not necessarily preserve $\psi$-mixing.

Examples demonstrate the theoretical results and their practical implications.

Abstract

This paper brings some insights of $\psi'$-mixing, $\psi^*$-mixing and $\psi$-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for $\psi$-mixing or $\psi'$-mixing, and also show that perturbations of $\psi'$-mixing copula-based Markov chains are $\psi'$-mixing while perturbations of $\psi$-mixing Markov chains are not necessarily $\psi$-mixing Markov chains, even when the perturbed copula is $\psi-mixing$. Some examples of copula families are considered. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov chains. Moreover, we provide a correction to a statement made in Longla and al. (2021) on $\psi$-mixing.