Strong Laws of Large Numbers for Products of Random Matrices

Abstract

This work, on products of random matrices, is inspired by papers of Furstenberg and Kesten (Ann. Math. Statist. 31 (1960), 457-469) and Furstenberg (Trans. Amer. Math. Soc. 108 (1963), 377-428). In particular, a formula was known for almost sure limits for normalized products of random matrices in terms of a stationary measure. However, no explicit computational techniques were known for these limits, and little was known about the stationary measures. We prove two main theorems. The first assumes that the random matrices are upper triangular and computes the almost sure limits in question. For the second, we assume the random matrices are $2 \times 2$ and Bernoulli, i.e., random matrices whose support is two points. Then the second theorem gives an asymptotic result for the almost sure limits, with rates of convergence in some cases.