We consider three matrix models of order 2 with one random entry $ \epsilon $ and the other three entries being deterministic. In the first model, we let $ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $ \epsilon\sim \rm{Bernoulli}\left(p\right) $ and $ p\in [0, 1] $ is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with $\xi\epsilon$ where $\epsilon$ is a standard Cauchy random variable and $\xi$ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

中文翻译：

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与随机斐波那契序列有关的随机矩阵乘积的Lyapunov指数和方差

我们考虑阶数为2的三个矩阵模型，其中一个随机项$ \ epsilon $，而其他三个项是确定性的。在第一个模型中，我们让$ \ epsilon \ sim \ rm {Bernoulli} \ left（\ frac {1} {2} \ right）$。对于此模型，我们开发了一种新技术，可以根据涉及斐波那契式序列的多级递归获得最高Lyapunov指数的估计值。反过来，就这些序列而言，这为Lyapunov指数提供了新的表征。在第二个模型中，当$ \ epsilon \ sim \ rm {Bernoulli} \ left（p \ right）$和[0，1] $中的$ p \作为参数时，我们给出相似的估计。这两个模型都与随机斐波那契序列有关。在最后一个模型中，当随机条目替换为$ \ xi \ epsilon $时，我们精确计算Lyapunov指数，其中$ \ epsilon $是标准柯西随机变量，而$ \ xi $是实参。然后，我们使用蒙特卡洛模拟对两个参数模型的CLT中的方差进行近似。