Random Fibonacci sequences and the number $1.13198824\dots$

Abstract

\begin{abstract} For the familiar Fibonacci sequence --defined by $f_1 = f_2 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n<2$ --$f_n$ increases exponentially with $n$ at a rate given by the golden ratio $(1+\sqrt{5})/2=1.61803398\ldots$. But for a simple modification with both additions and subtractions --the {\it random} Fibonacci sequences defined by $t_1=t_2=1$, and for $n<2$, $t_n = \pm t_{n-1} \pm t_{n-2}$, where each $\pm$ sign is independent and either $+$ or $-$ with probability $1/2$ --it is not even obvious if $\abs{t_n}$ should increase with $n$. Our main result is that \begin{equation*} \sqrt[n]{\abs{t_n}} \rightarrow 1.13198824\ldots\:\:\: \text{as}\:\:\: n \rightarrow\infty \end{equation*} with probability $1$. Finding the number $1.13198824\ldots$ involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal-like measure, a computer calculation, and a rounding error analysis to validate the computer calculation. \end{abstract}