Some limit properties for the mmth-order nonhomogeneous Markov chains indexed by an mm rooted Cayley tree

In this paper, we first study a convergence theorem for a finite mmth-order nonhomogeneous Markov chain indexed by an mm rooted Cayley tree. As corollaries, we obtain some limit theorems for the frequencies of occurrence of states for this Markov chain. Finally, we obtain the strong law of large numbers (LLN) and the Shannon–McMillan theorem for a class of finite mmth-order nonhomogeneous Markov chains indexed by an mm rooted Cayley tree.