The asymptotic equipartition property for a special Markov random field

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 study the generalized asymptotic equipartition property in the form of moving average for N bifurcating Markov chains indexed by an N-branch Cayley tree, which is a special case of Markov Urandom fields. Firstly, we construct a class of random variables containing a parameter with means of 1, and establish a strong limit theorem for the moving average of multivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we present the strong law of large numbers for the frequencies of occurrence of states of the moving average, as well as the generalized asymptotic equipartition property for N bifurcating Markov chains indexed by an N-branch Cayley tree. As corollaries, we also generalize some known results.