Some Limit Properties of Random Transition Probability for Second-Order Nonhomogeneous Markov Chains Indexed by a Tree

Abstract

We study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the property of the harmonic mean of random transition probability for a nonhomogeneous Markov chain.

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 graph distance to be the number of edges contained in the path .

Let be an arbitrary infinite tree that is partially finite (i.e., it has infinite vertices, and each vertex connects with finite vertices) and has a root . Meanwhile, we consider another kind of double root tree ; that is, it is formed with the root of connecting with an arbitrary point denoted by the root . For a better explanation of the double root tree , we take Cayley tree for example. It is a special case of the tree , the root of Cayley tree has neighbors, and all the other vertices of it have neighbors each. The double root tree (see Figure 1) is formed with root of tree connecting with another root .

Figure 1

Double root tree .

Let , be vertices of the double root tree . Write if is on the unique path connecting to , and for the number of edges on this path. For any two vertices , of the tree , denote by the vertex farthest from satisfying and .

The set of all vertices with distance from root is called the th generation of , which is denoted by . We say that is the set of all vertices on level and especially root is on the st level on tree . We denote by the subtree of the tree containing the vertices from level (the root ) to level and denote by the subtree of the tree containing the vertices from level (the root ) to level . Let be a vertex of the tree . We denote the first predecessor of by , the second predecessor of by , and denote by the th predecessor of . Let , and let be a realization of and denote by the number of vertices of .

Definition 1.1.

Let and be nonnegative functions on . Let

(1.1)

If

(1.2)

then is called a second-order transition matrix.

Definition 1.2.

Let be double root tree and let be a finite state space, and let be a collection of -valued random variables defined on the probability space . Let

(1.3)

be a distribution on , and

(1.4)

be a collection of second-order transition matrices. For any vertex t (), if

(1.5)

then is called a -value second-order nonhomogeneous Markov chain indexed by a tree with the initial distribution (1.3) and second-order transition matrices (1.4), or called a -indexed second-order nonhomogeneous Markov chain.

Remark 1.3.

Benjamini and Peres [1] have given the definition of the tree-indexed homogeneous Markov chains. Here we improve their definition and give the definition of the tree-indexed second-order nonhomogeneous Markov chains in a similar way. We also give the following definition (Definition 2.3) of tree-indexed nonhomogeneous Markov chains.

There have been some works on limit theorems for tree-indexed stochastic processes. Benjamini and Peres [1] have given the notion of the tree-indexed Markov chains and studied the recurrence and ray-recurrence for them. Berger and Ye [2] have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Ye and Berger (see [3, 4]), by using Pemantle's result [5] and a combinatorial approach, have studied the Shannon-McMillan theorem with convergence in probability for a PPG-invariant and ergodic random field on a homogeneous tree. Yang and Liu [6] have studied a strong law of large numbers for the frequency of occurrence of states for Markov chains field on a homogeneous tree (a particular case of tree-indexed Markov chains field and PPG-invariant random fields). Yang (see [7]) has studied the strong law of large numbers for frequency of occurrence of state and Shannon-McMillan theorem for homogeneous Markov chains indexed by a homogeneous tree. Recently, Yang (see [8]) has studied the strong law of large numbers and Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Huang and Yang (see [9]) have also studied the strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree.

Let . Then is called the random transition probability of a -indexed second-order nonhomogeneous Markov chain. Liu [10] has studied a strong limit theorem for the harmonic mean of the random transition probability of finite nonhomogeneous Markov chains. In this paper, we study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the results of [10, 11].

2. Main Results

Lemma 2.1.

Let be a -indexed second-order nonhomogeneous Markov chain with state space defined as in Definition 1.2, and let be a collection of functions defined on . Let , , and . Set

(2.1)

where is a real number. Then is a nonnegative martingale.

Proof.

Obviously, when , we have

(2.2)

Hence

(2.3)

Then

(2.4)

On the other hand, we also have

(2.5)

Combining (2.4) and (2.5), we arrive at

(2.6)

Thus the lemma is proved.

Theorem 2.2.

Let be a -indexed second-order nonhomogeneous Markov chain with state space defined as in Definition 1.2, and its initial distribution and probability transition collection satisfying

(2.7)

respectively. Let

(2.8)

If there exists such that

(2.9)

then the harmonic mean of the random conditional probability converges to a.e., that is,

(2.10)

Proof.

Let in Lemma 2.1. Then it follows from Lemma 2.1 that

(2.11)

is a nonnegative martingale. According to Doob martingale convergence theorem, we have

(2.12)

Thus

(2.13)

It follows from (2.11) and (2.13) that

(2.14)

By (2.14) and the inequalities , and , we have

(2.15)

It is easy to see that

(2.16)

Let , by (2.15), (2.16), (2.8), and (2.9) we have

(2.17)

Letting , by (2.17), we have

(2.18)

Let , by (2.15),(2.8), and (2.9) we have

(2.19)

Letting , by (2.19), we have

(2.20)

Combining (2.18) and (2.20), we obtain (2.10) directly.

From the definition above, we know that the difference between and lies in whether the root is connected with another root . In the following, we will investigate some properties of the harmonic mean of the transition probability of nonhomogeneous Markov chains on the tree . First, we give the definition of nonhomogeneous Markov chains on the tree .

Definition 2.3.

Let be an arbitrary tree that is partly finite, let be a finite state space, and let be a collection of -valued random variables defined on the probability space . Let

(2.21)

be a distribution on , and

(2.22)

be a collection of transition matrices. For any vertex (), if

(2.23)

then is called a -value nonhomogeneous Markov chain indexed by a tree with the initial distribution (2.21) and transition matrices (2.22), or called a -indexed nonhomogeneous Markov chain.

Let . Then is called the random transition probability of a -indexed nonhomogeneous Markov chain. Since a Markov chain is a special case of a second-order Markov chain, we may regard the nonhomogeneous Markov chain on to be a special case of the second-order nonhomogeneous Markov chain on when we do not take the difference of and on the root into consideration. Thus for the nonhomogeneous Markov chain on the tree , we can get the results similar to Lemma 2.1 and Theorem 2.2.

Lemma 2.4.

Let be a -indexed second-order nonhomogeneous Markov chain with state space defined as in Definition 2.3, and let be a collection of functions defined on . Let and . Set

(2.24)

where is a real number. Then is a nonnegative martingale.

Theorem 2.5.

Let be a -indexed nonhomogeneous Markov chain with state space defined as in Definition 2.3, and its initial distribution and probability transition collection satisfying

(2.25)

respectively. Let

(2.26)

If there exists such that

(2.27)

then the harmonic mean of the random conditional probability converges to a.e., that is

(2.28)

If the successor of each vertex of the tree has only one vertex, then the nonhomogeneous Markov chains on the tree degenerate into the general nonhomogeneous Markov chains. Thus we obtain the results in [10, 11].

Corollary 2.6 (see [10, 11]).

Let be a nonhomogeneous Markov chain with state space , and its initial distribution and probability transition sequence satisfying

(2.29)

respectively. Let

(2.30)

If there exists such that

(2.31)

then

(2.32)

Proof.

When the successor of each vertex of the tree has only one vertex, the nonhomogeneous Markov chains on the tree degenerate into the general nonhomogeneous Markov chains, the corollary follows directly from Theorem 2.5.