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

Abstract

In this paper, we first study a convergence theorem for a finite mth-order nonhomogeneous Markov chain indexed by an m 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 mth-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree.m rooted Cayley tree mth-order nonhomogeneous Markov chain Strong law of large numbers Shannon-McMillan theorem