Professor Ion Grama*,* Universite Bretagne Sud, Lorient (UBS)

**Title**: Conditioned limit theorems for products of random matrices and Markov chains with applications to branching processes.

**Abstrac**t: Consider a random walk defined by the consecutive action of independent identically distributed random matrices on a starting point outside the unit ball in the d dimensional Euclidean space. We study the first moment when the walk enters the unit ball. We study the exact behaviour of this time and prove conditioned limit theorems for the associated Markov walk. This extends to the case of walks on group GL(d,R) the well known results by Spitzer. The existence of the harmonic function related to the Markov walk turns out to be crucial point of the proof. We have extended these results to general Markov chains and applied them to study the branching processes in Markov environment.