Shortest positive products of nonnegative matrices

Let A={A1,…,Am} be a set of nonnegative d×d matrices having at least one strictly positive product (all products with no ordering and with repetitions permitted). What is the minimal possible length of their positive product? In other words, what is the minimal number l(A) for which there are indices i1,…,il(A)∈{1,…,m} such that the matrix Ai1…Ail(A) has all entries positive? In this paper we show, under some mild assumptions on matrices, that . We apply this result to estimate the rate of convergence in some limit theorems for random matrices.