Semigroups in semisimple groups

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.

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