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 epsilon and the other three entries being deterministic. In the first model, we let epsilon similar to Bernoulli (1/2). 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 similar to Bernoulli (p) and p is an element of [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.