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 ${\tilde{F}}_{n+2}=|\lambda {\tilde{F}}_{n+1}\pm {\tilde{F}}_n|$ (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 ${\tilde{F}}_n$ for $1/k<p\le 1$, is almost surely positive and given by $${\int }_0^{\infty }\log xd{\nu }_{k,\rho }(x),$$ 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 $p$.

References in corpus (3)

• Random Fibonacci Sequences
• How do random Fibonacci sequences grow?
• Growth rate for the expected value of a generalized random Fibonacci sequence

Cited by in corpus (1)

• Relative growth of the partial sums of certain random Fibonacci-like sequences