Two asymptotic distributions related to R\'enyi-type continued fraction expansions

We attempt to investigate a two-dimensional Gauss-Kuzmin theorem for R'enyi-type continued fraction expansions. More precisely speaking, our focus is to obtain specific lower and upper bounds for the error term considered which imply the convergence rate of the distribution function involved to its limit. To achieve our goal, we exploit the significant properties of the Perron-Frobenius operator of the R'enyi-type map under its invariant measure on the Banach space of functions of bounded variation. Finally, we give some numerical calculations to conclude the paper.