In this talk, an extension of the notion of big images and preimages (or 'finite primitivity' in the terminology of Mauldin & Urbanski) is extended to random topological Markov chains. It then turns out that this combinatorical condition is a sufficient condition for the existence of a random conformal measure, a random eigenfunction of the Ruelle operator and, moreover, implies that the system is relatively exact. In particular, these consequences suggest that these systems are the analogue of deterministic systems with the Gibbs-Markov property.

These results extend the work of Bogenschuetz, Gundlach and Kifer to random shift spaces with countably many states, and as an application one obtains a partial solution to a question of Orey on the convergence to the stationary distribution for Markov chains in random environment.