Strong Law of Large Numbers for Countable Markov Chains Indexed by an Infinite Tree with Uniformly Bounded Degree

A tree T is a graph which is connected and contains no circuits. Given any two vertices α ̸ = β ∈ T. Let αβ be the unique path connecting α and β. Define the distance d(α, β) to be the number of edges contained in the path αβ Select a vertex as the root (denoted by o). For any two vertices σ and t of tree T, we write σ ≤ t if σ is on the unique path from the root o to t. We denote by σ ∧ t the vertex farthest from o satisfying σ ∧ t ≤ t and σ ∧ t ≤ σ. For any vertex t of tree T, we denote by |t| the distance between o and t. The set of all vertices with distance n from the root o is called the nth level of T. For any vertex t of tree T, we denote the predecessor of t by 1 t , the predecessor of 1 t by 2 t , and the predecessor of (n − 1) t by n t . We also call n t the nth predecessor of t. Similarly, we denote the one of the successor of t by 1, the one of the successors of 1t by 2, and one of the successors of (n − 1)t by n. We denote by T(n) the subtree comprised of level 0 (the root o) through level n, and byL n the set of all vertices on level n. 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 M neighboring vertices and other vertices have M + 1 neighboring vertices, we call this type of tree a Cayley tree and denote it by T C,M . It is easy to see that this type of tree is the special case of uniformly bounded tree. Let S be the subgraph of T, XS = {X t , t ∈ S}, and xS the realization of X. We denote by |S| the number of vertices of S.


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 1  , the predecessor of 1  by 2  , and the predecessor of ( − 1)  by   .We also call   the th predecessor of .Similarly, we denote the one of the successor of  by 1  , the one of the successors of 1  by 2  , and one of the successors of ( − 1)  by   .We denote by  () the subtree comprised of level 0 (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  + 1 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  = {0, 1, 2, . ..} be a countable state space and {  ,  ∈ } a collection of -valued random variables defined on the probability space (Ω, F, P).
be a distribution on , and let be a stochastic matrix on 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 .
The convergence of   () to a constant in a sense ( 1 convergence, convergence in probability, and a.e.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  1 and a.e.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 treeindexed Markov chains and studied the recurrence and rayrecurrence 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 PPG-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.

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  2 .Let Then for all  ≥ 0, we have Proof.Huang  ), as  ∈   ,  ≥ 1, ∀ ∈ , ∀ℎ ≥ 1, and we have Proof.We only need to prove the situation of ℎ = 2.By the Markov property (3), we have By induction, (10) holds for ℎ ≥ 2.

Strong Law of Large Numbers
In the following, let  ≥ 0,  ∈ ,  0 () = 1, let   be the th predecessor of  defined as before, and where Proof.Let   (, ) =   ()  () for all  ∈  in Lemma 3; then by ( 8) and ( 12), we have +1 () P ( Since  is a uniformly bounded tree, so {  (, ) =   ()  (),  ∈ } are uniformly bounded functions defined on  2 ; then, from Lemma 3, we have lim Let   (, ) =  +1 ()P( 1  =  |   = ) in Lemma 3; by Definition 1 and Lemma 4, we have Since {  (, ) =  +1 ()P( 1  =  |   = ),  ∈ } are also uniformly bounded functions defined on  2 , from Lemma 3 and (18), for any  ≥ 0, we have By ( 12), we have Since  is strongly ergodic, the first term of right-hand side of ( 25) is arbitrary small for large ℎ, and the limit of second term is zero as  → ∞; (15) can be obtained from ( 22) and (25).Let  = 0 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.