  This paper continues our investigation of Renyi-type continued fractions
studied in \cite{Sebe&Lascu-2018}. A Wirsing-type approach to the
Perron-Frobenius operator of the R\'enyi-type continued fraction transformation
under its invariant measure allows us to study the optimality of the
convergence rate. Actually, we obtain upper and lower bounds of the convergence
rate which provide a near-optimal solution to the Gauss-Kuzmin-L\'evy problem.
