Stable Laws and Products of Positive Random Matrices

Let S be the multiplicative semigroup of q×q matrices with positive entries such that every row and every column contains a strictly positive element. Denote by (X n ) n≥1 a sequence of independent identically distributed random variables in S and by X (n)=X n ⋅⋅⋅ X 1, n≥1, the associated left random walk on S. We assume that (X n ) n≥1 satisfies the contraction property $$\mathbb {P}\biggl(\bigcup_{n\geq1}[X^{(n)}\in S{^{\circ}}]\biggr)>0,$$ where S° is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random matrix X 1 which ensure that the logarithms of the entries, of the norm, and of the spectral radius of the products X (n), n≥1, are in the domain of attraction of a stable law.