Lyapunov exponent for the products of random matrices related to an art collector model

We consider a special case of an art collector model recently proposed by Rastegar, Roitershtein, Roytershteyn and Seetharam (2024). The model considers an investor collector who must balance the competing long-term goals of sustainable financial health and maintaining a pleasing art collection. The model studies the exponential growth rate of the long-term value of the assets through the products of random matrices where one of the entries involves a Bernoulli random variable with parameter . This exponential growth rate is known as the Lyapunov exponent. Exact values for Lyapunov exponents are generally difficult to compute, even for simple random matrix models. We give a novel construction of a deterministic sequence of upper and lower bounds that converge to the Lyapunov exponent. This sequence involves a recursion in two variables related to Fibonacci type sequences.