A Two-Dimensional Gauss–Kuzmin Theorem Associated with the Random Fibonacci-Type Sequences

Abstract

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 $$\vartheta _l=\frac{2(l-1)}{2l-1+\sqrt{1+4(l-1)}}$$ .