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.

This minicourse is an introduction to the group of conformal transformations of the sphere. We will focus on its action on the space of non-singleton compact subsets of the sphere.
1. Conformal autosimilarity: we will see that the compact subsets  whose orbit is closed are exactly the limit sets of convex cocompact subgroups.
2. Quasicircles: we  will see that the compact subsets whose orbit closure contains only Jordan curves are exactly the quasicircles.
3. Minimal orbits: we will construct compact subsets of the sphere whose orbit is not closed but whose orbit closure is minimal.