Convergence in distribution of products of random matrices

Abstract

We consider a sequence A2, A2, ... of i.i.d. nonnegative matrices of size d × d, and investigate convergence in distribution of the product Mn: =A1 ... An. When d≧2 it is possible for Mn to converge in distribution (without normalization) to a distribution not concentrated on the zero matrix. Several equivalent conditions for this to happen are given. These lead to a fairly general family of examples. These conditions can also be used to determine when the a.s. limit of 1/nlog∥Mn∥ equals the logarithm of the largest eigenvalue of E(A1). © 1984 Springer-Verlag.