Limit theorems for products of positive random matrices

Let S be the set of q x q matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by S degrees the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence (X-n)(n greater than or equal to 1) in S. The aim of this paper is to describe the asymptotic behavior of the random products X-(n) = X-n... X-1, n greater than or equal to 1, under the main hypothesis P(U)(n greater than or equal to 1)[X-(n) is an element of S degrees]) > 0. We first study the behavior "in direction" of row and column vectors of X-(n). Then, adding a moment condition, we prove a law of large numbers for the entries and lengths of these Vectors and also for the spectral radius of X-(n). Under the mixing hypotheses that are usual in the case of sums of real random variables, we get a central limit theorem for the previous quantities. The variance of the Gaussian limit law is strictly positive except when (X-(n))(n greater than or equal to 1) is tight. This tightness property is fully studied when the X-n, n greater than or equal to 1, are independent.