Exponential growth of random infinite Fibonacci sequences

TL;DR

This paper proves that random infinite Fibonacci sequences grow exponentially almost surely, with the growth rate characterized by a Lyapunov exponent, answering a question posed in prior research.

Contribution

It establishes the almost sure exponential growth rate of a class of random Fibonacci-like sequences and characterizes this rate via a Lyapunov exponent, resolving an open question.

Findings

Sequences grow exponentially almost surely

Growth rate is given by a positive Lyapunov exponent

Answers a previously open question in the literature

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}).