Random matrix products when the top Lyapunov exponent is simple

28 pages, 1 figureIn the present paper, we treat random matrix products on the general linear group $GL(V)$, where $V$ is a vector space defined on any local field, when the top Lyapunov exponent is simple, without irreducibility assumption. In particular, we show the existence and uniqueness of the stationary measure $\nu$ on $P(V)$ that is relative to the top Lyapunov exponent and we describe the projective subspace generated by its support. Then, we relate this support to the limit set of the semi-group $T_{\mu}$ of $GL(V)$ generated by the random walk. Moreover, we show that $\nu$ has H\"older regularity and give some limit theorems concerning the behavior of the random walks: exponential convergence in direction, large deviation estimates of the probability of hitting an hyperplane. These results generalize known ones when $T_{\mu}$ acts strongly irreducibly and proximally (i-p to abbreviate) on $V$. In particular, when applied to the affine group in the so-called contracting case, the H\"older regularity of the stationary measure together with the description of the limit set are new. We mention that we don't use results from the i-p setting; rather we see it as a particular case