A Gauss-Kuzmin-L\'evy theorem for R\'enyi-type continued fractions

We consider an interval map which is a generalization of the R'enyi transformation. For the continued fraction expansion arising from this transformation, we prove a result concerning the asymptotic behavior of the distribution functions of this map. More exactly, we use Sz"usz’s method to prove a Gauss-Kuzmin-L'evy-type theorem.