Shannon-McMillan theorem and strong law of large numbers for Markov chains indexed by generalized spherically symmetric trees

In this paper, we investigate some strong limit properties for Markov chains indexed by a class of infinite trees, which are the generalizations of the spherically symmetric tree. We first give the definition of the generalized spherically symmetric tree model, then obtain the strong law of large numbers for Markov chain indexed by such tree. Finally, we establish the Shannon-McMillan theorem for Markov chains indexed by generalized spherically symmetric tree by applying the Stolz theorem. The obtained properties are generalizations of some known results in the existing literature.