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

Abstract

<p style='text-indent:20px;'>We consider three matrix models of order 2 with one random entry <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the other three entries being deterministic. In the first model, we let <inline-formula><tex-math id="M2">\begin{document}$ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $\end{document}</tex-math></inline-formula>. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when <inline-formula><tex-math id="M3">\begin{document}$ \epsilon\sim \rm{Bernoulli}\left(p\right) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p\in [0, 1] $\end{document}</tex-math></inline-formula> is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with <inline-formula><tex-math id="M5">\begin{document}$\xi\epsilon$\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M6">\begin{document}$\epsilon$\end{document}</tex-math></inline-formula> is a standard Cauchy random variable and <inline-formula><tex-math id="M7">\begin{document}$\xi$\end{document}</tex-math></inline-formula> is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.