Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

We consider three matrix models of order 2 with one random entry $ϵ$ and the other three entries being deterministic. In the first model, we let $ϵ\sim \text{Bernoulli}\left(\frac{1}{2}\right)$. 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 $ϵ\sim \text{Bernoulli}\left(p\right)$ 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 $\xi ϵ$ where $ϵ$ 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.