  In this paper, we study dependence coefficients for copula-based Markov
chains. We provide new tools to check the convergence rates of mixing
coefficients of copula-based Markov chains. We study Markov chains generated by
the Metropolis-hastings algorithm and give conditions on the proposal that
ensure exponential $\rho$-mixing, $\beta$-mixing and $\phi$-mixing. A general
necessary condition on symmetric copulas to generate exponential $\rho$-mixing
or $\phi$-mixing is given. At the end of the paper, we comment and improve some
of our previous results on mixtures of copulas.
