Exponential growth of random infinite Fibonacci sequences

Abstract

We consider the recursion $X_{n+1}={\sum }_{i=0}^n{\epsilon }_{n,i}{X}_{n-i}$, where ${\epsilon }_{n,i}$ are i.i.d. (Bernoulli) random variables taking values in $\{-1,1\}$, and $X_0=1$, $X_{-j}=0$ for $j>0$. We prove that almost surely, $n^{-1}\log |X_n|\to \bar{\gamma }>0$, where $\bar{\gamma }$ is an appropriate Lyapunov exponent. This answers a question of Viswanath and Trefethen (\textit{SIAM J. Matrix Anal. Appl. 19:564--581, 1998}).

Minor revision, fixed proof in appendix