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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$\epsilon\sim\textrm{Bernoulli}\left(\frac{1}{2}\right)$. For this model wedevelop a new technique to obtain estimates for the top Lyapunov exponent interms of a multi-level recursion involving Fibonacci-like sequences. This inturn gives a new characterization for the Lyapunov exponent in terms of thesesequences. In the second model, we give similar estimates when$\epsilon\sim\textrm{Bernoulli}\left(p\right)$ and $p\in [0,1]$ is a parameter.Both of these models are related to random Fibonacci sequences. In the lastmodel, we compute the Lyapunov exponent exactly when the random entry isreplaced with $\xi\epsilon$ where $\epsilon$ is a standard Cauchy randomvariable and $\xi$ is a real parameter. We then use Monte Carlo simulations toapproximate the variance in the CLT for both parameter models.

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