Some strong limit theorems for Markov chain fields on trees

In this article, we introduce the notion of the Markov chain fields on the generalized Bethe trees or generalized Cayley trees, and some strong limit theorems on the frequencies of states and ordered couples of states, including the Shannon-McMillan theorem on Bethe tree T-B,T-N and Cayley tree T-C,T-N, are obtained. In the proof, a new technique in the study of the strong limit theorem in probability theory is applied.