Limit theorems for products of positive random matrices

Let S be the set of q × q matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by S^∘ the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence (Xₙ)n ≥1 in S. The aim of this paper is to describe the asymptotic behavior of the random products X⁽ⁿ⁾ =Xₙ … X ₁, n≥ 1 under the main hypothesis P(n≥ 1[X⁽ⁿ⁾∈ S^∘])>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≥ 1 is tight. This tightness property is fully studied when the Xₙ, n≥ 1, are independent.