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

TL;DR

This paper introduces new techniques to estimate Lyapunov exponents for specific random matrix models related to Fibonacci sequences, and analyzes the variance in the CLT for these models through simulations.

Contribution

It develops a novel recursive method to estimate Lyapunov exponents and provides exact and approximate results for different random matrix models related to Fibonacci sequences.

Findings

New recursive estimates for Lyapunov exponents in matrix models.

Exact Lyapunov exponent calculation for Cauchy-distributed entries.

Monte Carlo simulations for variance approximation in CLT.

Abstract

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 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 $\epsilon\sim\textrm{Bernoulli}\left(p\right)$ and $p\in [0,1]$ is a parameter. Both of these models are related to random Fibonacci sequences. In the 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.