On Some Mixing Properties of Copula-Based Markov Chains

Abstract

AbstractThis paper brings some insights of $$\psi '$$ ψ ′ -mixing, $$\psi ^*$$ ψ ∗ -mixing and $$\psi$$ ψ -mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for $$\psi$$ ψ -mixing or $$\psi '$$ ψ ′ -mixing. We show that perturbations of $$\psi '$$ ψ ′ -mixing copula-based Markov chains are $$\psi '$$ ψ ′ -mixing while perturbations of $$\psi$$ ψ -mixing Markov chains are not necessarily $$\psi$$ ψ -mixing Markov chains, even when the perturbed copula generates $$\psi$$ ψ -mixing. The Farlie–Gumbel–Morgenstern, gaussian and Ali-Mikhail-Haq copula families are considered among other examples. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov chains in a simulation study. Moreover, we provide a correction to a statement made in Longla et al. (J Korean Stat Soc, 1–23, 2021) on $$\psi$$ ψ -mixing.