Chan (, ) considered some continued fraction expansions related to random Fibonacci-type sequences. A Wirsing-type approach to the Perron-Frobenius operator of the associated 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.
"Convergence Rate for a Continued Fraction Expansion Related to Fibonacci Type Sequences." Tokyo J. Math. 33 (2) 487 - 497, December 2010. https://doi.org/10.3836/tjm/1296483483