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In the present paper, we treat random matrix products on the general lineargroup $\textrm{GL}(V)$, where $V$ is a vector space defined on any local field,when the top Lyapunov exponent is simple, without irreducibility assumption. Inparticular, we show the existence and uniqueness of the stationary measure$\nu$ on $\textrm{P}(V)$ that is relative to the top Lyapunov exponent and wedescribe the projective subspace generated by its support. We observe that thedynamics takes place in a open set of $\textrm{P}(V)$ which has the structureof a skew product space. Then, we relate this support to the limit set of thesemi-group $T_{\mu}$ of $\textrm{GL}(V)$ generated by the random walk.Moreover, we show that $\nu$ has H\"older regularity and give some limittheorems concerning the behavior of the random walk and the probability ofhitting a hyperplane. These results generalize known ones when $T_{\mu}$ actsstrongly irreducibly and proximally (i-p to abbreviate) on $V$. In particular,when applied to the affine group in the so-called contracting case or moregenerally when the Zariski closure of $T_{\mu}$ is not necessarily reductive,the H\"older regularity of the stationary measure together with the descriptionof the limit set are new. We mention that we don't use results from the i-psetting; rather we see it as a particular case.

Random matrix products when the top Lyapunov exponent is simple