On Dependence Structure of Copula-based Markov chains

We consider dependence coefficients for stationary Markov chains. We emphasize on some equivalencies for reversible Markov chains. We improve some known results and provide a necessary condition for Markov chains based on Archimedean copulas to be exponential $\rho$-mixing. We analyze the example of the Mardia and Frechet copula families using small sets.