  Via dependence with complete connections we investigate the $\psi$-mixing
coefficients of the sequence $(a_n)_{n \in \mathbb{N}}$ of incomplete quotients
and also of the doubly infinite sequence $(\overline{a}_l)_{l \in \mathbb{Z}}$
of extended incomplete quotients of the R\'enyi-type continued fraction
expansions. A L\'evy-type approach allows us to obtain good upper bounds for
these coefficients.
