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arXiv · 2505.00377

Exponential growth of random infinite Fibonacci sequences

Ilya Goldsheid·Ofer Zeitouni
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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}).

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Ilya Goldsheid, Ofer Zeitouni. 2025-05-01. Exponential growth of random infinite Fibonacci sequences. https://arxiv.org/abs/2505.00377

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