The AEP for Hidden Markov Tree Models

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.