Strong law of large numbers and Shannon-McMillan theorem for Markov chain fields on trees

Abstract

We study the strong law of large numbers and the Shannon-McMillan theorem for Markov chain fields on trees. First, we prove the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for Markov chain fields on trees. Then, we prove the Shannon-McMillan theorem with almost everywhere (a.e.) convergence for Markov chain fields on trees. We prove the results on a Bethe tree and then just state the analogous results on a rooted Cayley tree. In the proof, a new technique for establishing the strong limit theorem in probability theory is applied.