In this article, we attempt to investigate a two-dimensional Gauss–Kuzmin theorem for continued fraction expansions associated with random Fibonacci-type sequences introduced by Chan (2006). More precisely speaking, our focus is to obtain a Gauss–Kuzmin theorem concerning the natural extension of corresponding interval maps \(\{\tau _l: l\in \mathbb {N},l\ge 2\}\). Then, we give a local betterment of this theorem. It matters that together with characteristic properties of the Perron–Frobenius operator of \(\tau _l\) under its invariant measure on the Banach space of functions of bounded variation, we are in a position to conclude unambiguous lower and upper bounds for the error term linked to distribution function in the case when \(2\le l\le 257\), which show that the desired optimal convergence rate is \(\mathcal {O}(\vartheta _l^n)\) as \(n\rightarrow \infty \) with \(\vartheta _l=\frac{2(l-1)}{2l-1+\sqrt{1+4(l-1)}}\).

中文翻译：

---

与随机斐波那契型序列相关的二维高斯-库兹明定理

在本文中，我们尝试研究二维Gauss-Kuzmin定理，以解决由Chan（2006）引入的与随机斐波那契类型序列相关的连续分数扩展。更确切地说，我们的重点是获得有关相应区间图\（\ {\ tau _l：l \ in \ mathbb {N}，l \ ge 2 \} \）的自然扩展的Gauss-Kuzmin定理。然后，我们对该定理进行局部改进。重要的是，结合\（\ tau _l \）的Perron–Frobenius算子在有界变化函数的Banach空间上的不变测度的特征性质，我们可以得出该误差的上下界在\（2 \ le l \ le 257 \）的情况下，术语与分配函数相关联，这表明期望的最佳收敛速度是\（\ mathcal {O}（\ vartheta _l ^ n）\）作为\（n \ rightarrow \ infty \），其中\（\ vartheta _l = \ frac {2（l- 1）} {2l-1 + \ sqrt {1 + 4（l-1）}} \）。