This paper continues our investigation of Rényi-type continued fractions studied in Lascu and Sebe (A dependence with complete connections approach to generalized Rényi continued fractions, Acta Math. Hungar. 160, 292–313, 2020 ). A Wirsing-type approach to the Perron–Frobenius operator of the Rényi-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–Lévy problem.

中文翻译：

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Rényi 型连分数展开的收敛率

本文继续我们对 Lascu 和 Sebe 中研究的 Rényi 型连分数的研究（Adependency with complete connections approach to generalized Rényi 连分数，Acta Math. Hungar. 160, 292–313, 2020）。Rényi 型连分数变换的 Perron-Frobenius 算子在其不变测度下的 Wirsing 型方法使我们能够研究收敛速度的最优性。实际上，我们获得了收敛速度的上限和下限，这为 Gauss-Kuzmin-Lévy 问题提供了近乎最优的解决方案。