An elementary proof that random Fibonacci sequences grow exponentially

Eran Makover; Jeffrey McGowan

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An elementary proof that random Fibonacci sequences grow exponentially

Eran Makover·Jeffrey McGowan

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Abstract

We consider random Fibonacci sequences given by $x_{n+1}=\pm βx_{n}+x_{n-1}$. Viswanath (\cite{viswanath}), following Furstenberg (\cite{furst}) showed that when $β= 1$, $\lim_{n\to \infty}|x_{n}|^{1/n}=1.13...$, but his proof involves the use of floating point computer calculations. We give a completely elementary proof that $1.25577 \ge (E(|x_{n}|))^{1/n} \ge 1.12095$ where $E(|x_{n}|)$ is the expected value for the absolute value of the $n$th term in a random Fibonacci sequence. We compute this expected value using recurrence relations which bound the sum of all possible $n$th terms for such sequences. In addition, we give upper an lower

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BibTeX RIS

Eran Makover, Jeffrey McGowan. 2005-10-28. An elementary proof that random Fibonacci sequences grow exponentially. https://arxiv.org/abs/math/0510159

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