The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we shall study the strong law of large numbers and the AEP for hidden Markov tree models. First, we give a strict definition of hidden Markov tree models, and study their key properties and equivalent characterizations. In fact, hidden Markov tree models are closely related to the tree-indexed Markov chains, therefore, we also introduce the definition of tree-indexed Markov chains and their equivalent properties. We also establish a strong law of large numbers for hidden Markov tree models indexed by a Cayley tree. As corollaries, we obtain some strong laws of large numbers for the parameters and the AEP for these models.

Guyon (Guyon J. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann Appl Probab, 2007, 17: 1538-1569) introduced an important model for homogeneous bifurcating Markov chains indexed by a binary tree taking values in general state space and studied their limit theorems. The results were applied to detect cellular aging. In this paper, we define a discrete form of nonhomogeneous bifurcating Markov chains indexed by a binary tree and discuss the equivalent properties for them. The strong law of large numbers and the entropy ergodic theorem are studied for these Markov chains with finite state space. In contrast to previous work, we use a new approach to prove the main results of this paper.