  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.
