This paper brings some insights of ψ′-mixing, ψ∗-mixing and ψ-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for ψ-mixing or ψ′-mixing. We show that perturbations of ψ′-mixing copula-based Markov chains are ψ′-mixing while perturbations of ψ-mixing Markov chains are not necessarily ψ-mixing Markov chains, even when the perturbed copula generates 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 ψ-mixing.
Longla, M., Mous-Abou, H., & Ngongo, I. S. (2022). On some mixing properties of copula-based markov chains. Journal of Statistical Theory and Applications, 21(3), 131–154. https://doi.org/10.1007/s44199-022-00045-1