Limit theorems for asymptotic circular $m$th-order Markov chains indexed by an $m$-rooted homogeneous tree

Abstract

In this paper, we  give the definition of an asymptotic circular $m$th-order Markov chain indexed by an $m$ rooted homogeneous tree.  By applying the limit property for a sequence of multi-variables functions of a nonhomogeneous Markov chain indexed by such tree, we establish the strong law of large numbers and the asymptotic equipartition property (AEP) for asymptotic circular $m$th-order finite Markov chains indexed by this homogeneous tree.  As a corollary, we can obtain the strong law of large numbers and AEP about the $m$th-order finite nonhomogeneous Markov chain indexed by the $m$ rooted homogeneous tree.