LYAPUNOV EXPONENT AND VARIANCE IN THE CLT FOR PRODUCTS OF RANDOM MATRICES RELATED TO RANDOM FIBONACCI SEQUENCES

摘要

We consider three matrix models of order 2 with one random entry e and the other three entries being deterministic. In the first model, we let (∈) ～ 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 (∈) ～ Bernoulli (p) and p ∈ [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 ξ∈ where ∈ is a standard Cauchy random variable and ξ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.