The Generalized Entropy Ergodic Theorem for Nonhomogeneous Markov Chains Indexed by a Homogeneous Tree

Abstract

In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).

An analogue of topological sequence entropy for Markov hom tree-shifts

“…This line of research traces its origin back to [7], in which the authors studied stationary random fields on rooted/unrooted d-trees. Subsequent works, such as [18,9], have also made progress in this field. Noteworthy contributions regarding deviation inequalities are also provided by [13], [14] and [17].…”

Small deviation properties concerning arrays of non-homogeneous Markov information sources

“…From the 1970s to the early stages of the 21st century, the AEP for various general stochastic processes was investigated by many studies, such as [8][9][10][11][12][13]. Recently, many scholars, such as Yang (e.g., [14][15][16][17]), Shi (e.g., [3,[18][19][20][21]), Huang [22,23], and Peng [24][25][26], by generalizing the method proposed by [27], [11], and Wang [2,28,29], studied the AEP and the limit properties (including AEP and SLLNs) of some types of Markov chains (such as homogeneous and non-homogeneous; finite state space and infinite state space; and Markov chains indexed by the set of positive integers and tree-indexed Markov chains).…”

A Class of Small Deviation Theorems for Functionals of Random Fields on a Tree With Uniformly Bounded Degree in Random Environment

“…Peng [17] has given the strong law of large numbers for Markov chains indexed by spherically symmetric trees. Huang [9] has studied the generalized entropy theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Shi et al [23] have given the equivalent properties for the bifurcating Markov chains indexed by a binary tree.…”