Strong laws of large numbers for asymptotic even-odd Markov chains indexed by a homogeneous tree

In this paper, we are going to study the strong laws of large numbers for asymptotic even-odd Markov chains indexed by a homogeneous tree. First, the definition of the asymptotic even-odd Markov chain is introduced. Then the strong limit theorem for asymptotic even-odd Markov chains indexed by a homogeneous tree is established. Next, the strong laws of large numbers for the frequencies of occurrence of states and ordered couple of states for asymptotic even-odd Markov chains indexed by a homogeneous tree are obtained. Finally, we prove the asymptotic equipartition property (AEP) for these Markov chains.