Exponential growth of random infinite Fibonacci sequences
Ilya Goldsheid, Ofer Zeitouni
TL;DR
The paper analyzes the almost-sure exponential growth rate of the Bernoulli random infinite Fibonacci recursion $X_{n+1}=\sum_{i=0}^n\epsilon_{n,i}X_{n-i}$ with $X_0=1$, proving the existence of a positive Lyapunov exponent $\bar{\gamma}$ such that $\frac{1}{n}\log|X_n|\to\bar{\gamma}$ a.s. and $\frac{1}{n}\log\|\hat{X}_n\|_2\to\bar{\gamma}$. The method recasts the process as products of random operators $A_n=U+K_n$ on a Hilbert space and leverages the Goldsheid–Margulis framework (via Kingman’s theorem) to obtain the Lyapunov exponent, while handling boundedness with the $H_{c,2}$ spaces. The authors also establish the existence and positivity of a second Lyapunov limit for a broad class of initial data and distributions, and discuss extensions beyond Bernoulli, including a Gaussian-special-case contraction argument in Appendix B. This work resolves an open question of Viswanath & Trefethen and provides a robust operator-theoretic approach to exponential growth in random linear recursions with memory.
Abstract
We consider the recursion $X_{n+1}=\sum_{i=0}^n ε_{n,i}X_{n-i}$, where $ε_{n,i}$ 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, $n^{-1}\log |X_n|\to \bar γ>0$, where $\bar γ$ is an appropriate Lyapunov exponent. This answers a question of Viswanath and Trefethen (\textit{SIAM J. Matrix Anal. Appl. 19:564--581, 1998}).