Let $S$ be the multiplicative semigroup of $q\times q$ matrices with positive entries such that every row and every column contains a strictly positive element. Denote by $(X_n)_{ n\geq1}$ a sequence of independent identically distributed random variables in $S$ and by $X^{ (n)} = X_n ... X_1$, $ n\geq 1$, the associated left random walk on $S$. We assume that $(X_n)_{ n\geq1}$ verifies the contraction property $\P(\bigcup_{ n\geq1}[X^{ (n)} \in S^\circ])>0$, where $S^\circ $ 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 $X^{ (n)}$, $n\ge 1$, are in the domain of attraction of a stable law.
Categories: math.PR math.FA