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Exponential growth of random infinite Fibonacci sequences
Ilya Goldsheid, Ofer Zeitouni · 2025-05-01 · via math.PR updates on arXiv.org

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}).

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