  Dependence coefficients have been widely studied for Markov processes defined
by a set of transition probabilities and an initial distribution. This work
clarifies some aspects of the theory of dependence structure of Markov chains
generated by copulas that are useful in time series econometrics and other
applied fields. The main aim of this paper is to clarify the relationship
between the notions of geometric ergodicity and geometric {\rho}-mixing;
namely, to point out that for a large number of well known copulas, such as
Clayton, Gumbel or Student, these notions are equivalent. Some of the results
published in the last years appear to be redundant if one takes into account
this fact. We apply this equivalence to show that any mixture of Clayton,
Gumbel or Student copulas generate both geometrically ergodic and geometric
{\rho}-mixing stationary Markov chains, answering in this way an open question
in the literature. We shall also point out that a sufficient condition for
{\rho}-mixing, used in the literature, actually implies Doeblin recurrence.
