Let d > 1. Let A be a random matrix taking values in one of the following (semi-)groups:
- GL(d,R), the group of invertible real matrices
- R> x O(d), the group of similarity matrices
- M(dxd,R>=), the semi-group of nonnegative matrices
Goal: To study properties of the left random walk (=product of matrices)
Pin = An *... * A1,
where (An) is a sequence of i.i.d.copies of A; and to study its action on Rd.
I will describe the history of the problem, explain what assumptions are needed to prove limit theorems for Pin and highlight recent research directions; including conditional limit theorems and approximative duality.