On mixtures of copulas and mixing coefficients

We show that if the density of the absolutely continuous part of a copula is bounded away from zero on a set of Lebesgue measure 1, then that copula generates \textquotedblleft lower $\psi$-mixing\textquotedblright\ stationary Markov chains. This conclusion implies $\phi$-mixing, $\rho$-mixing, $\beta$-mixing and \textquotedblleft interlaced $\rho$-mixing\textquotedblright . We also provide some new results on the mixing structure of Markov chains generated by mixtures of copulas.