Random Fibonacci sequences and the number 1.13198824…

Abstract

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