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.