We will first explain three properties of a Zariski dense subsemigroups of a semisimple real Lie group: - the existence of loxodromic elements, - the convexity and non degeneracy of the limit cone, - the density of the group spanned by the Jordan projection. We will then explain how to deduce from them three limit theorems for a random walk on this semigroup: - the law of large numbers, - the central limit theorem, - the local limit theorem.