Convergence in distribution of products of random matrices

Summary We consider a sequence A₂, A₂, ... of i.i.d. nonnegative matrices of size d × d, and investigate convergence in distribution of the product M_n: =A₁ ... A_n. When d2 it is possible for M_n 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/nlogM_n equals the logarithm of the largest eigenvalue of E(A₁).