Convergence rate for R\'enyi-type continued fraction expansions

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.