Almost-sure Growth Rate of Generalized Random Fibonacci sequences

Abstract

We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\ge 1$, $F_{n+2} = \lambda F_{n+1} \pm F_{n}$ (linear case) and $\widetilde F_{n+2} = |\lambda \widetilde F_{n+1} \pm \widetilde F_{n}|$ (non-linear case), where each $\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\le 1$). Our main result is that, when $\lambda$ is of the form $\lambda_k = 2\cos (\pi/k)$ for some integer $k\ge 3$, the exponential growth of $F_n$ for $0<p\le 1$, and of $\widetilde F_{n}$ for $1/k < p\le 1$, is almost surely positive and given by $$ \int_0^\infty \log x d\nu_{k, \rho} (x), $$ where $\rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $\nu_{k, \rho}$ is an explicit probability distribution on $\RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<p\le 1$ in the easier case $\lambda\ge 2$. Finally, we study the variations of the exponent as a function of $p$.