Limit theorems for products of positive random matrices

Abstract

Let $S$ be the set of $q \times q$ matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by $S^\circ$ the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence $(X_n)_{n \geq1}$ in $S$. The aim of this paper is to describe the asymptotic behavior of the random products $X^{(n)} =X_n \ldots X _1, n\geq 1$ under the main hypothesis $P(\Bigcup_{n\geq 1}[X^{(n)}\in S^\circ])>0$. 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 $(X^{(n)})_{n\geq 1}$ is tight. This tightness property is fully studied when the $X_n, n\geq 1$, are independent.