Some limit properties for Markov chains indexed by a homogeneous tree

We first study a local convergence theorem for countable Markov chains indexed by a homogeneous tree. As corollaries, we obtain some limit theorems for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a homogeneous tree. Finally, we obtain the strong law of large numbers and Shannon-McMillan theorem for finite Markov chains indexed by a homogeneous tree. In the proof, a new technique for establishing the strong limit theorems in probability theory is applied.