Abstract
We study the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by an infinite tree with uniformly bounded degree, which extends the corresponding results of countable Markov chains indexed by a Cayley tree and generalizes the relative results of finite Markov chains indexed by a uniformly bounded tree.
1. Introduction
A tree is a graph which is connected and contains no circuits. Given any two vertices . Let be the unique path connecting and . Define the distance to be the number of edges contained in the path
Select a vertex as the root (denoted by ). For any two vertices and of tree , we write if is on the unique path from the root to . We denote by the vertex farthest from satisfying and . For any vertex of tree , we denote by the distance between and . The set of all vertices with distance from the root is called the th level of . For any vertex of tree , we denote the predecessor of by , the predecessor of by , and the predecessor of by . We also call the th predecessor of . Similarly, we denote the one of the successor of by , the one of the successors of by , and one of the successors of by . We denote by the subtree comprised of level (the root ) through level , and by the set of all vertices on level . In this paper, we mainly consider an infinite tree which has uniformly bounded degrees. That is, the numbers of neighbors of any vertices in this tree are uniformly bounded; we call it the uniformly bounded tree. If the root of a tree has neighboring vertices and other vertices have neighboring vertices, we call this type of tree a Cayley tree and denote it by . It is easy to see that this type of tree is the special case of uniformly bounded tree. Let be the subgraph of , , and the realization of . We denote by the number of vertices of .
Definition 1 (see [1]). Let be a local finite and infinite tree; that is, the tree has infinite vertices and the degrees of any vertices in this tree are finite. Let be a countable state space and a collection of -valued random variables defined on the probability space . Let be a distribution on , and let be a stochastic matrix on . Let be any positive integer. If for any vertices , , , will be called -valued Markov chains indexed by this tree with the initial distribution (1) and the transition matrix (2) or called -indexed Markov chains with state space .
Definition 2 (see [2, page 157]). Let be a stochastic matrix defined on the countable state space . If there exists a distribution satisfying where is the -step transition probability determined by , then is said to be strongly ergodic with distribution . Obviously, if (5) holds, then we have , and is said to be the stationary distribution determined by .
Let be a sequence of random variables taking values in state space with the joint distribution Let is called the entropy density of .
The convergence of to a constant in a sense ( convergence, convergence in probability, and . convergence) is called Shannon-McMillan-Breiman theorem or asymptotic equipartition property (AEP) in information theory. Shannon [3] first proved AEP in convergence in probability for finite stationary ergodic sequence of random variables. McMillan [4] and Breiman [5, 6] proved AEP in and . convergence, respectively, for finite stationary ergodic sequence of random variables. Chung [7] generalized Breiman's result to countable case.
The subject of tree-indexed processes is rather young. Benjamini and Peres [1] have given the notion of the tree-indexed Markov chains and studied the recurrence and ray-recurrence for them. Guyon [8] has given the definition of bifurcating Markov chains indexed by binary tree and studied their limit theorems. Berger and Ye [9] have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Ye and Berger [10, 11] by using Pemantle's result [12] and a combinatorial approach have studied the asymptotic equipartition property (AEP) in the sense of convergence in probability for a -invariant and ergodic random field on a homogeneous tree. Yang [13] has studied the strong law of large numbers and the asymptotic equipartition property (AEP) for finite Markov chains indexed by a homogeneous tree. Yang and Ye [14] have studied the strong law of large numbers and the asymptotic equipartition property (AEP) for finite level-nonhomogeneous Markov chains indexed by a homogeneous tree. Huang and Yang [15] have studied the strong law of large numbers and the asymptotic equipartition property (AEP) for finite Markov chains indexed by an infinite tree with uniformly bounded degree. Recently, Wang et al. [16] have studied the strong law of large numbers for countable Markov chains indexed by a Cayley tree.
In some previous articles, only the tree-indexed Markov chains with the finite state space are considered; meanwhile the countable case has very important theoretical significance, so Chung [7] generalized Breiman's result [5, 6] to the countable case. Wang et al. [16] have studied the strong law of large numbers countable Markov chains indexed by a Cayley tree.
The technique used to study the strong law of large numbers for countable Markov chains indexed by trees is different from that for finite case. The processing method of finite state space cannot apply to countable state space, because the sum and limit cannot be exchanged. For studying the strong law of large numbers for countable Markov chains indexed by trees, we first establish a strong limit theorem then use this strong limit theorem and smoothing property of conditional expectation repeatedly to establish our strong law of large numbers. In this paper, we use the same approach used in [16] to study the strong law of large numbers for Markov chains indexed by a uniformly bounded tree. Our results generalize the results of Huang and Yang [15] for finite Markov chains indexed by a uniformly bounded tree (Figure 1) and the results of Wang et al. [16] for countable Markov chains indexed by a Cayley tree.
2. Some Lemmas
Before proving the main results, we begin with some lemmas.
Lemma 3. Let be an infinite tree with uniformly bounded degree, a -indexed Markov chain with countable state space defined as before, and uniformly bounded functions defined on . Let Then for all , we have
Proof. Huang and Yang (see [15, Theorem 1]) have obtained a similar result for finite Markov chains indexed by a uniformly bounded tree. By checking carefully the proof of that theorem, one can find it also holds for countable Markov chains indexed by a uniformly bounded tree, so the proof of this lemma is omitted.
Lemma 4. Let and be defined as Lemma 3, , as , , , , and we have
Proof. We only need to prove the situation of . By the Markov property (3), we have By induction, (10) holds for .
3. Strong Law of Large Numbers
In the following, let , , , let be the th predecessor of defined as before, and where
Theorem 5. Let be an infinite tree with uniformly bounded degree, a strongly ergodic stochastic matrix, and the unique stationary distribution of . Let be a -indexed Markov chain taking values in countable state space with the stochastic matrix . Then, for any integer , we have
Proof. Let for all in Lemma 3; then by (8) and (12), we have Since is a uniformly bounded tree, so are uniformly bounded functions defined on ; then, from Lemma 3, we have Let in Lemma 3; by Definition 1 and Lemma 4, we have Since are also uniformly bounded functions defined on , from Lemma 3 and (18), for any , we have Hence, By (17) and (20), we have By induction, for fixed and all , we have By (12), we have As , we have as Since is strongly ergodic, the first term of right-hand side of (25) is arbitrary small for large , and the limit of second term is zero as ; (15) can be obtained from (22) and (25).
Theorem 6. Under the conditions of Theorem 5, let be defined by (13); then
Proof. Let in Lemma 3; it is easy to see that are uniformly functions defined in , and, by (8) and Lemma 3, we have Equation (26) follows from (28) and Theorem 5.
Let in Theorems 5 and 6; we can obtain the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by the uniformly bounded tree.
Corollary 7. Under the conditions of Theorem 5, let then
Proof. Letting in Theorems 5 and 6, this corollary follows.
From Theorems 5 and 6, we can obtain easily the strong law of large numbers for countable Markov chains indexed by a Cayley tree [16] and the strong law of large numbers for finite Markov chains indexed by a uniformly bounded tree [15].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China 11071104, the Postgraduate Innovation Projection of Jiangsu University CXLX12-0652, Youth Foundation of Xuzhou Institute of Technology XKY2012301, National Natural Science Foundation of China 11226210, the Research Foundation for Advanced Talents of Jiangsu University 11JDG116, and the National Natural Science Foundation of China 11326174.