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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.00377 |
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| _version_ | 1866910924473892864 |
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| author | Goldsheid, Ilya Zeitouni, Ofer |
| author_facet | Goldsheid, Ilya Zeitouni, Ofer |
| contents | 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}). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential growth of random infinite Fibonacci sequences Goldsheid, Ilya Zeitouni, Ofer Probability 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}). |
| title | Exponential growth of random infinite Fibonacci sequences |
| topic | Probability |
| url | https://arxiv.org/abs/2505.00377 |