随机Fibonacci序列指数增长的一个初等证明
An elementary proof that random Fibonacci sequences grow exponentially
摘要：
考虑由$x_{n+1}=\pm βx_{n}+x_{n-1}$给出的随机Fibonacci序列。Viswanath（\cite{viswanath}）继Furstenberg（\cite{furst}）之后证明了$β= 1$、$\lim_{n\to \infty}|x_{n}|^{1/n}=1.13...$，但他的证明涉及到利用计算机进行浮点计算。本文给出了随机Fibonacci数列中$E(|x_{n}|)$第项绝对值的期望值为$1.25577 \ge (E(|x_{n}|))^{1/n} \ge 1.12095$的一个完全初等的证明。我们使用递归关系计算这个期望值，该递归关系限定了这些序列的所有可能的$n$ th项的和。此外，我们给出了上一个下一个
英文摘要：
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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