A class of small deviation theorems for the random fields on an m rooted Cayley tree

Abstract

In this paper, we are to establish a class of strong deviation theorems for the random fields relative to m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMillan theorem for m th-order nonhomogeneous Markov chains indexed by that tree.

2000 Mathematics Subject Classification: 60F15; 60J10.

1. Introduction

A tree is a graph G = {T, E} which is connected and contains no circuits. Given any two vertices σ, t(σ ≠ t ∈ T), let $\overline{\sigma t}$ be the unique path connecting σ and t. Define the graph distance d (σ, t) to be the number of edges contained in the path $\overline{\sigma t}$.

Let T _C,N be a Cayley tree. In this tree, the root (denoted by o) has only N neighbors and all other vertices have N + 1 neighbors. Let T _{B, N} be a Bethe tree, on which each vertex has N + 1 neighboring vertices. Here both T _C,N and T _B,N are homogeneous tree. In this paper, we mainly consider an m rooted Cayley tree ${\overline{T}}_{C,N}$ (see Figure 1). It is formed by a Cayley tree T _C,N with the root o connecting with another vertex denoted by the the root -1, and then root -1 connecting with another vertex denoted by the root -2, and continuing to do the same work until the last vertex denoted by the root - (m - 1) is connected. When the context permits, this type of tree is denoted simply by T.

Figure 1

An m rooted Cayley tree ${\bar{T}}_{C,2}$.

Let σ, t(σ, t ≠ o, -1, - 2,..., - (m - 1)) be vertices of an m rooted Cayler tree T. Write t ≤ σ if t is on the unique path connecting o to σ, and |σ | the number of edges on this path. For any two vertices σ, t(σ, t ≠ o, -1, - 2,..., - (m - 1)) of tree T, denote by σ ∧ t the vertex farthest from o satisfying σ ∧ t ≤ σ and σ ∧ t ≤ t.

The set of all vertices with distance n from the root o is called the n-th generation of T, which is denoted by L _n . We say that L _n is the set of all vertices on level n and especially root -1 is on the -1st level on tree T, root -2 is on the -2nd level. By analogy, root -(m - 1) is on the -(m - 1) th level. We denote by T⁽ⁿ⁾the subtree of an m rooted Cayley tree T containing the vertices from level -(m - 1) (the root -(m - 1)) to level n. Let t(t ≠ o, -1, -2, ..., -(m - 1)) be a vertex of an m rooted Cayley tree T. Predecessor of the vertex t is another vertex, which is nearest from t, on the unique path from root -(m - 1) to 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 say that n _t is the n-th predecessor of t. X^A = {X _t , t ∈ A} is a stochastic process indexed by a set A, and denoted by |A| the number of vertices of A, x^A is the realization of X^A.

Let $\left(\mathrm{\Omega },\mathcal{F}\right)$ be a measure space, {X _t , t∈T} be a collection of random variables defined on $\left(\mathrm{\Omega },\mathcal{F}\right)$ and taking values in G = {0,1,..., b - 1}, where b is a positive integer. Let P be a general probability distribution on $\left(\mathrm{\Omega },\mathcal{F}\right)$. We will call P the random field on tree T. Denote the distribution of {X _t , t ∈ T} under the probability measure P by

$$P\left(x^{T^{\left(n\right)}}\right)=P\left(X^{T^{\left(n\right)}}=x^{T^{\left(n\right)}}\right),x^{T^{\left(n\right)}}\in G^{T^{\left(n\right)}}.$$

(1)

Let

$$f_n\left(\omega \right)=-\frac{1}{\left|T^{\left(n\right)}\right|}\text{ln}P\left(X^{T^{\left(n\right)}}\right).$$

(2)

f _n (ω) is called entropy density of $X^{T^{\left(n\right)}}$.

Let Q be another probability measure on the measurable space $\left(\mathrm{\Omega },\mathcal{F}\right)$, and let the distribution of {X _t , t ∈ T} under Q be

$$Q\left(x^{T^{\left(n\right)}}\right)=Q\left(X^{T^{\left(n\right)}}=x^{T^{\left(n\right)}}\right),x^{T^{\left(n\right)}}\in G^{T^{\left(n\right)}}.$$

(3)

Let

$$h\left(P|Q\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\text{ln}\frac{P\left(X^{T^{\left(n\right)}}\right)}{Q\left({X^T}^{\left(n\right)}\right)}.$$

(4)

h(P | Q) is called the sample divergence rate of P relative to Q.

Remark 1 If P = Q, h(P | Q) = 0 holds. By using the approach of Lemma 1 of Liu and Wang [1], we also can prove that h(P | Q) ≥ 0, P - a.e.; hence, h(P | Q) can be regarded as a measure of the Markov approximation of the arbitrary random field on T.

Definition 1 (see [2]) Let G = {0, 1,..., b - 1} and P(y|x₁, x₂,..., x _m ) be a nonnegative functions on G^m+1. Let

$$P=\left(P\left(y|x_1,x_2,\dots ,x_m\right)\right),P\left(y|x_1,x_2,\dots ,x_m\right)\ge 0,x_1,x_2,\dots ,x_m,y\in G.$$

If

$$\sum _{y\in G}P\left(y|x_1,x_2,\dots ,x_m\right)=1,$$

then P is called an m-order transition matrix.

Definition 2 (see [2]). Let T be an m rooted Cayley tree, and let G = {0, 1,..., b - 1} be a finite state space, {X _t , t ∈ T} be a collection of G-valued random variables defined on the probability space $\left(\mathrm{\Omega },\mathcal{F},Q\right)$. Let Q be a probability on a measurable space $\left(\mathrm{\Omega },\mathcal{F}\right)$.

Let

$$q=\left(q\left(x_1,x_2,\dots ,x_m\right)\right),x_1,x_2,\dots ,x_m\in G$$

(5)

be a distribution on G^m, and

$$Q_n=\left(q_n\left(y|x_1,x_2,\dots ,x_m\right)\right),x_1,x_2,\dots ,x_m,y\in G,n\ge 1$$

(6)

be m-order transition matrices. For any vertex t ∈ L _n , n ≥ 1, if

$$\begin{array}{c}Q\left(X_t=y|X_{1_t}=x_1,X_{2_t}=x_2,\dots ,X_{m_t}=x_m\mathsf{\text{ and }}{X}_{\sigma }\mathsf{\text{ for}}\sigma \wedge t\le 1_t\right)\\ =Q\left(X_t=y|X_{1_t}=x_1,X_{2_t}=x_2,\dots ,X_{m_t}=x_m\right)\\ =q_n\left(y|x_1,x_2,\dots ,x_m\right),\forall x_1,x_2,\dots ,x_m,y\in G\end{array}$$

(7)

and

$$\begin{array}{c}Q\left(X_{-\left(m-1\right)}=x_1,\dots ,X_{-1}=x_{m-1},X_o=x_m\right)\\ =q\left(x_1,\dots ,x_{m-1},x_m\right),x_1,\dots ,x_m\in G,\end{array}$$

(8)

then {X _t , t ∈ T} is called a G-valued m th-order nonhomogeneous Markov chain indexed by an m rooted Cayley tree with the initial m dimensional distribution (5) and m-order transition matrices (6) under the probability measure Q, or called a T-indexed m th-order nonhomogeneous Markov chain under the probability measure Q.

We denote

$$o_m=\left\{o,-1,-2,\dots ,-\left(m-1\right)\right\},o_m^{\prime }=\left\{-1,-2,\dots ,-\left(m-1\right)\right\},$$

$$X_1^n\left(t\right)=\left\{X_{n_t},\dots ,X_{2_t},X_{1_t}\right\},X_0^n\left(t\right)=\left\{X_{n_t},\cdot \cdot \cdot ,X_{2_t},X_{1_t},X_t\right\},$$

and denote by $x_1^n\left(t\right)$ and $x_0^n\left(t\right)$ the realizations $X_1^n\left(t\right)$ and $X_0^n\left(t\right)$, respectively.

Let {X _t , t ∈ T} be an m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree T under the probability measure Q defined on above. It is easy to see that

$$Q\left(x^{T^{\left(n\right)}}\right)=Q\left(X^{T^{\left(n\right)}}=x^{T^{\left(n\right)}}\right)=q\left(x_{-\left(m-1\right)},\dots ,x_o\right)\prod _{k=1}^n\prod _{t\in L_k}{q}_k\left(x_t|x_1^m\left(t\right)\right).$$

(9)

In the following, we always assume that P(x^T(ⁿ)), Q(x^T(ⁿ)), q(x₁,..., x _m ), and {q _n (y | x₁,..., x _m ), n ≥ 1} are all positive.

There have been some works on limit theorems for tree-indexed stochastic process. Benjamini and Peres [3] have given the notion of the tree-indexed Markov chains and studied the recurrence and ray-recurrence for them. Berger and Ye [4] have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Pemantle [5] proved a mixing property and a weak law of large numbers for a PPG-invariant and ergodic random field on a homogeneous tree. Ye and Berger [6, 7], by using Pemantle's result 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 [8] have studied a strong law of large numbers for the frequency of occurrence of states for Markov chains field on a Bethe tree (a particular case of tree-indexed Markov chains field and PPG-invariant random field). Yang [9] 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. Yang and Ye [10] have studied the strong law of large numbers and Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Huang and Yang [11] have studied the strong law of large numbers and Shannon-McMillan theorem for Markov chains indexed by an infinite tree with uniformly bounded degree. Recently, Shi and Yang [12] have also studied some limit properties of random transition probability for second-order nonhomogeneous Markov chains indexed by a tree. Peng et al. [13] have studied a class of strong deviation theorems for the random fields relative to homogeneous Markov chains indexed by a homogeneous tree. Shi and Yang [2] have studied the strong law of large numbers and Shannon-McMillan for the m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. Yang [14] has also studied a class of small deviation theorems for the sequences of N-valued random variables with respect to m th-order nonhomogeneous Markov chains.

In this paper, our main purpose is to extend Yang's [14] result to an m rooted Cayley tree. By introducing the sample divergence rate of any probability measure with respect to m th-order nonhomogeneous Markov measure on an m rooted Cayley tree, we establish a class of strong deviation theorems for the arbitrary random fields indexed by that tree with respect to m th-order nonhomogeneous Markov chains indexed by that tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMillan theorem for m th-order nonhomogeneous Markov chains indexed by that tree.

2. Main Results

Before giving the main results, we begin with a lemma.

Lemma 1 Let T be an m rooted Cayley tree, G = {0, 1,..., b - 1} be the finite state space. Let {X _t , t ∈ T} be a collection of G-valued random variables defined on the measurable space $\left(\mathrm{\Omega },\mathcal{F}\right)$. Let P and Q be two probability measures on the measurable space $\left(\mathrm{\Omega },\mathcal{F}\right)$, and let {X _t , t ∈ T} be an m th-order nonhomogeneous Markov chains indexed by tree T under probability measure Q. Let {g _n (y₁,..., y _{m +1}), n ≥ 1} be a sequence of functions defined on G^m+1. Let ${ℱ}_n=\sigma (X^{T^{(n)}})(n\ge 1)$. Set

$$F_n\left(\omega \right)=\sum _{k=1}^n\sum _{t\in L_k}{g}_k\left(X_0^m\left(t\right)\right)$$

(10)

and

$$t_n\left(\lambda ,\omega \right)=\frac{e^{\lambda F_n\left(\omega \right)}}{{\prod }_{k=1}^n{\prod }_{t\in L_k}{E}_Q\left[e^{\lambda g_k\left(X_0^m\left(t\right)\right)}|X_1^m\left(t\right)\right]}\cdot \frac{q\left(X_{-\left(m-1\right)},\dots ,X_o\right){\prod }_{k=1}^n{\prod }_{t\in L_k}{q}_k\left(X_t|X_1^m\left(t\right)\right)}{P\left(x^{T^{\left(n\right)}}\right)},$$

(11)

where E _Q denote the expectation under probability measure Q. Then $\left\{t_n\left(\lambda ,\omega \right),{\mathcal{F}}_n,n\ge 1\right\}$ is a nonnegative martingale under probability measure P.

Proof The proof is similar to Lemma 3 of Peng et al. [12], so the proof is omitted.

Theorem 1 Let T be an m rooted Cayley tree, {X _t , t ∈ T} be a collection of random variables taking values in G = {0, 1,..., b - 1} defined on the measurable space $\left(\mathrm{\Omega },\mathcal{F}\right)$. Let P and Q be two probability measures on the measurable space $\left(\mathrm{\Omega },\mathcal{F}\right)$, such that {X _t , t ∈ T} is an m th-order nonhomogeneous Markov chain indexed by T under Q. Let h(P | Q) be defined by (4), {g _n (y₁,..., y _{m +1}), n ≥ 1} be a sequence of functions defined on G^m+1. Let c ≥ 0 be a constant. Set

$$D\left(c\right)=\left\{\omega :h\left(P|Q\right)\le c\right\}.$$

(12)

Assume that there exists α > 0, such that ∀i^m ∈ G^m,

$$b_{\alpha }\left(i^m\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}{E}_Q\left[e^{a\left|g_k\left(X_0^m\left(t\right)\right)\right|}|X_1^m\left(t\right)=i^m\right]\le \tau .$$

(13)

Let

$$A_t=\frac{2\tau }{e^2{\left(t-\alpha \right)}^2},$$

(14)

where o < t < a. Thus, when 0 ≤ c ≤ t²A _t , we have

$$\underset{n\to \infty }{\underset{n\to \infty }{\lim \sup }\frac{1}{\left|T^{(n)}\right|}\left|{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}\{g_k(X_0^m(t))-E_Q[g_k(X_0^m(t))|X_1^m(t)]\}}}\right|\lim \sup }\le 2\sqrt{c{A}_t}\mathrm{,}P-a\mathrm{.}e\mathrm{.,}\omega \in D(c).$$

(15)

In particular,

$$\underset{n\to \infty }{\text{lim}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}=0,P-a.e.,\omega \in D\left(0\right).$$

(16)

Proof Let t _n (λ, ω) be defined by (11). By Lemma 1, $\left\{t_n\left(\lambda ,\omega \right),{\mathcal{F}}_n,n\ge 1\right\}$ is a non-negative martingale under probability measure P. By Doob's martingale convergence theorem, we have

$$\underset{n\to \infty }{\text{lim}}{t}_n\left(\lambda ,\omega \right)=t\left(\lambda ,\omega \right)<\infty ,P-a.e.$$

Hence,

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\text{ln}{t}_n\left(\lambda ,\omega \right)\le 0,P-a.e..$$

(17)

We have by (9), (10), (11) and (17)

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\left[\sum _{k=1}^n\sum _{t\in L_k}\left\{\lambda g_k\left(X_0^m\left(t\right)\right)-\text{ln}{E}_Q\left[e^{\lambda g_k\left(X_0^m\left(t\right)\right)}|X_1^m\left(t\right)\right]\right\}-\text{ln}\frac{P\left(X^{T^{\left(n\right)}}\right)}{Q\left(X^{T\left(n\right)}\right)}\right]\le 0,P-a.e.$$

(18)

By (4),(12) and (18)

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{\lambda g_k\left(X_0^m\left(t\right)\right)-\text{ln}{E}_Q\left[e^{\lambda g_k\left(X_0^m\left(t\right)\right)}|X_1^m\left(t\right)\right]\right\}\le c,P-a.e.,\omega \in D\left(c\right).$$

(19)

This implies that

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \le \underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{\text{ln}{E}_Q\left[e^{\lambda g_k\left(X_0^m\left(t\right)\right)}|X_1^m\left(t\right)\right]-E_Q\left[\lambda g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}+c,P-a.e.,\omega \in D\left(c\right)\end{array}$$

(20)

Let |λ| < t. By inequalities In x ≤ x -1(x > 0) and $e^x-1-x\le \frac{x^2}{2}{e}^{\left|x\right|}$, and noticing that

$$\text{max}\left\{x^2{e}^{-hx},x\ge 0\right\}=4e^{-2}/h^2\left(h>0\right).$$

(21)

We have

$$\begin{array}{l}\underset{n\to \infty }{\lim \sup }\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}\left\{\ln E_Q\left[e^{\lambda g_k(X_0^m(t))}|X_1^m(t)\right]-E_Q[\lambda g_k(X_0^m(t))|X_1^m(t)]\right\}}}\\ \le \underset{n\to \infty }{\lim \sup }\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}\left\{E_Q\left[e^{\lambda g_k(X_0^m(t))}|X_1^m(t)\right]-1-E_Q[\lambda g_k(X_0^m(t))|X_1^m(t)]\right\}}}\\ \le \frac{{\lambda }^2}{2}\underset{n\to \infty }{\lim \sup }\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}{E}_Q\left[g_{k}2(X_0^m(t))e^{|\lambda {\left|\right|}_{g_k}(X_0^m(t))|}|X_1^m(t)\right]}}\\ =\frac{{\lambda }^2}{2}\underset{n\to \infty }{\lim \sup }\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}{E}_Q\left[e^{\alpha |g_k(X_0^m(t))|}{g}_{k}2(X_0^m(t))e^{(|\lambda |-\alpha )|g_k(X_0^m(t))|}|X_1^m(t)\right]}}\\ \le \frac{{\lambda }^2}{2}\underset{n\to \infty }{\lim \sup }\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}{E}_Q\left[e^{\alpha |g_k(X_0^m(t))|}4e^{-2}/{(|\lambda |-a)}^2|X_1^m(t)\right]}}\\ \le 2{\lambda }^2\tau /e^2{(t-\alpha )}^2.\end{array}$$

(22)

By (20) and (22), we have

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \le {\lambda }^2{A}_t+c,P-a.e.,\omega \in D\left(c\right).\end{array}$$

(23)

When 0 < λ < t < α, we have by (23)

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \le \lambda A_t+c/\lambda ,P-a.e.,\omega \in D\left(c\right).\end{array}$$

(24)

It is easy to see that when 0 < c < t²A _t , the function f (λ) = λA _t + c/λ attains, at $\lambda =\sqrt{c/A_t}$, its smallest value $f\left(\sqrt{c/A_t}\right)=2\sqrt{c{A}_t}$. Letting $\lambda =\sqrt{c/A_t}$ in (24), we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\le 2\sqrt{c{A}_t},P-a.e.,\omega \in D\left(c\right).$$

(25)

When c = 0, we have by (24)

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\le \lambda A_t,P-a.e.,\omega \in D\left(0\right).$$

(26)

Letting λ → 0⁺ in (26), we obtain

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\le 0,P-a.e.,\omega \in D\left(0\right).$$

(27)

Hence, (25) also holds for c = 0. When -α < -t < λ < 0, by virtue of (23) it can be shown in a similar way that

$$\underset{n\to \infty }{\text{lim}\text{inf}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\ge -2\sqrt{c{A}_t},P-a.e.,\omega \in D\left(c\right).$$

(28)

Equation 15 follows from (25) and (28), Equation 15 implies (16) immediately. This completes the proof of the theorem. □

Theorem 2 Let

$$H_t=2b/e^2{\left(t-1\right)}^2,0<t<1.$$

(29)

Let f _n (ω) be defined by (2). Under the conditions of Theorem 1, when 0 ≤ c ≤ t²H _t , we have

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\left\{f_n\left(\omega \right)-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right]\right\}\\ \le 2\sqrt{c{H}_t},P-a.e.,\omega \in D\left(c\right),\end{array}$$

(30)

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}\left\{f_n\left(\omega \right)-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right]\right\}\\ \ge -2\sqrt{c{H}_t}-c,P-a.e.,\omega \in D\left(c\right),\end{array}$$

(31)

where H(p₀,.... p _{b -1}) denote the entropy of distribution (p₀,..., p _{b -1}), i.e.,

$$H\left(p_0,\dots ,p_{b-1}\right)=-\sum _{i=0}^{b-1}{p}_i\text{ln}{p}_i.$$

Proof In Theorem 1, let g _k (y₁,..., y _{m +1}) = - In q _k (y _{m +1} | y₁,..., y _m ) and α = 1, we have

$$\begin{array}{c}{E}_Q\left[e^{g_k\left(X_0^m\left(t\right)\right)}|X_1^m\left(t\right)=i^m\right]\\ =\sum _{j\in G}{e}^{|-\text{ln}{q}_k\left(j|i^m\right)|}{q}_k\left(j|i^m\right)\\ =\sum _{j\in G}{q}_k\left(j|i^m\right)/q_k\left(j|i^m\right)\\ =b.\end{array}$$

(32)

Hence, ∀i^m ∈ G^m,

$$b_1\left(i^m\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}{E}_Q\left[e^{g_k\left(X_0^m\left(t\right)\right)}|X_1^m\left(t\right)=i^m\right]\le b.$$

(33)

Noticing that

$$\begin{array}{c}{E}_Q\left[-\text{ln}{q}_k\left(X_t|X_1^m\left(t\right)\right)|X_1^m\left(t\right)\right]\\ =-\sum _{j\in G}{q}_k\left(j|X_1^m\left(t\right)\right)\text{ln}{q}_k\left(j|X_1^m\left(t\right)\right)\\ =H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right].\end{array}$$

(34)

When 0 ≤ c ≤ t²H _t , we have by (34),(29) and (15)

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\left\{\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left(-\text{ln}{q}_k\left(X_t|X_1^m\left(t\right)\right)\right)-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right]\right\}\\ \le 2\sqrt{c{H}_t},P-a.e.,\omega \in D\left(c\right).\end{array}$$

(35)

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}\left\{\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left(-\text{ln}{q}_k\left(X_t|X_1^m\left(t\right)\right)\right)-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right]\right\}\\ \ge -2\sqrt{c{H}_t},P-a.e.,\omega \in D\left(c\right).\end{array}$$

(36)

By (35), (9) and h(P|Q) ≥ 0,

$$\begin{array}{l}\underset{n\to \infty }{\lim \sup }\{f_n(\omega )-\frac{1}{\left|T^{(n)}\right|}{\displaystyle \underset{k=1}{\overset{n}{|}}{\displaystyle \underset{t\in L_k}{|}H[q_k(0|X_1^m(t)),\dots \mathrm{,}{q}_k(b-1|X_1^m(t))]}}\\ \le \underset{n\to \infty }{\lim \sup }\{-\frac{1}{\left|T^{(n)}\right|}\ln P(X^{T^{(n)}})-\frac{1}{\left|T^{(n)}\right|}{\displaystyle \underset{k=1}{\overset{n}{|}}{\displaystyle \underset{t\in L_k}{|}(-\ln q_k(X_t|X_1^m(t))}}\}\\ +\underset{n\to \infty }{\lim \sup }\{\frac{1}{\left|T^{(n)}\right|}{\displaystyle \underset{k=1}{\overset{n}{|}}{\displaystyle \underset{t\in L_k}{|}(-\ln q_k(X_t|X_1^m(t))}}\\ -\frac{1}{\left|T^{(n)}\right|}{\displaystyle \underset{k=1}{\overset{n}{|}}{\displaystyle \underset{t\in L_k}{|}H[q_k(0|X_1^m(t)),\dots \mathrm{,}{q}_k(b-1|X_1^m(t))]}}\}\\ \le 2\sqrt{c{H}_t}\mathrm{,}P-a\mathrm{.}e\mathrm{.,}\omega \in D(c).\end{array}$$

(37)

By (36), (9) and (12), we have

$$\begin{array}{l}\underset{n\to \infty }{\lim \sup }\{f_n(\omega )-\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}H[q_k(0|X_1^m(t)),\dots \mathrm{,}{q}_k(b-1|X_1^m(t))]}}\\ \le \underset{n\to \infty }{\lim \sup }\{-\frac{1}{\left|T^{(n)}\right|}\ln P(X^{T^{(n)}})-\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}(-\ln q_k(X_t|X_1^m(t))}}\}\\ +\underset{n\to \infty }{\lim \sup }\{\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}(-\ln q_k(X_t|X_1^m(t))}}\\ -\frac{1}{\left|T^{(n)}\right|}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}H[q_k(0|X_1^m(t)),\dots \mathrm{,}{q}_k(b-1|X_1^m(t))]}}\}\\ \le 2\sqrt{c{H}_t}\mathrm{,}P-a\mathrm{.}e\mathrm{.,}\omega \in D(c).\end{array}$$

(38)

This completes the proof of this theorem. □

Corollary 1 Under the conditions of Theorem 2, we have

$$\underset{n\to \infty }{\text{lim}}\left\{f_n\left(\omega \right)-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right]\right\}=0,P-a.e.,\omega \in D\left(0\right).$$

(39)

If P << Q, then

$$\underset{n\to \infty }{\text{lim}}\left\{f_n\left(\omega \right)-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right]\right\}=0,P-a.e.$$

(40)

In particular, if P = Q,

$$\underset{n\to \infty }{\text{lim}}\left\{f_n\left(\omega \right)-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(b-1|X_1^m\left(t\right)\right)\right]\right\}=0,Q-a.e.$$

(41)

Proof Letting c = 0 in (30) and (31), Equation 39 follows. If P << Q, then h(P | Q) = 0, P - a.e.,(cf. see [15],P.121), i.e., P(D(0)) = 1. Hence, Equation 40 follows from (39). In particular, if P = Q, then h(P | Q) ≡ 0. Hence, (41) follows from (40). □

Theorem 3 Under the conditions of Theorem 1, if {g _n (y₁,.... y _{m +1}), n ≥ 1} is uniformly bounded, i.e., there exists M > 0 such that |g _n (y₁,..., y _{m +1})| ≤ M, then when c ≥ 0, we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\left|\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\right|\le M\left(c+2\sqrt{c}\right),P-a.e.,\omega \in D\left(c\right).$$

(42)

Proof By (20) and (12) and the formula in line 2 of (22), we have

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \le \underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}{E}_Q\left[e^{\lambda g_k\left(X_0^m\left(t\right)\right)}-1-\lambda g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\\ +cP-a.e.,\omega \in D\left(c\right).\end{array}$$

(43)

By the hypothesis of the theorem and the inequality e^x - 1 - x ≤ |x|(e^|x|- 1), we have

$$e^{\lambda g_k\left(X_0^m\left(t\right)\right)}-1-\lambda g_k\left(X_0^m\left(t\right)\right)\le \left|\lambda \right|M\left(e^{\left|\lambda \right|M}-1\right).$$

(44)

By (43) and (44)

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \le \left|\lambda \right|M\left(e^{\left|\lambda \right||M}-1\right)+c,P-a.e.,\omega \in D\left(c\right).\end{array}$$

(45)

When λ > 0, we have by (45)

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \le M\left(e^{\lambda M}-1\right)+c/\lambda ,P-a.e.,\omega \in D\left(c\right).\end{array}$$

(46)

Taking $\lambda =\frac{1}{M}\text{log}\left(1+\sqrt{c}\right)$, and using the inequality

$$\text{log}\left(1+\sqrt{c}\right)\ge \frac{\sqrt{c}}{1+\sqrt{c}},$$

(47)

we have when c > 0

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \le M\sqrt{c}+\frac{cM}{\text{log}\left(1+\sqrt{c}\right)}\\ \le M\left(2\sqrt{c}+c\right),P-a.e.,\omega \in D\left(c\right).\end{array}$$

(48)

When λ < 0, it follows from (45) that

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \ge -M\left(e^{\lambda M}-1\right)+c/\lambda P-a.e.,\omega \in D\left(c\right).\end{array}$$

(49)

Taking $\lambda =-\frac{1}{M}\text{log}\left(1+\sqrt{c}\right)$ in (49), and using (47), we have when c > 0

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g_k\left(X_0^m\left(t\right)\right)-E_Q\left[g_k\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\\ \ge -M\sqrt{c}-\frac{cM}{\text{log}\left(1+\sqrt{c}\right)}\\ \ge -M\left(2\sqrt{c}+c\right),P-a.e.,\omega \in D\left(c\right).\end{array}$$

(50)

In a similar way, it can be shown that (48) and (50) also hold when c = 0. By (48) and (50), we have (42) holds. This completes the proof of this theorem.□

Corollary 2 Under the conditions of Theorem 1, let g(y₁,..., y _{m +1}) be any function defined on G^m+1. Let M = max g(y₁,..., y _{m +1}). Then when c ≥ 0,

$$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{\left|T^{\left(n\right)}\right|}\left|\sum _{k=1}^n\sum _{t\in L_k}\left\{g\left(X_0^m\left(t\right)\right)-E_Q\left[g\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}\right|\le M\left(c+2\sqrt{c}\right),P-a.e.,\omega \in D\left(c\right).$$

(51)

Proof Letting g(y₁,..., y _{m +1}) = g _n (y₁,..., y _{m +1}), n ≥ 1 in Theorem 3, this corollary follows.

In the following, let $I_k\left(x\right)=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill x=k\hfill \\ \hfill 0\hfill & \hfill x\ne k\hfill \end{array}\right.$. Let $S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i_1,\dots ,i_m\right)$ be the number of (i₁,..., i _m ) in the collection of $\left\{X_0^{m-1}\left(t\right),t\in T^{\left(n\right)}\backslash o_m^{\prime }\right\}$, that is

$$S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i_1,\dots ,i_m\right)=\sum _{k=0}^n\sum _{t\in L_k}{I}_{i_1}\left(X_{{\left(m-1\right)}_t}\right)\cdot \cdot \cdot I_{i_m}\left(X_t\right),$$

(52)

$S_{T^{\left(n\right)}\backslash o_m}\left(i_1,\dots ,i_m,i_{m+1}\right)$ be the number of (i₁,..., i _m , i _{m +1}) in the collection of $\left\{X_0^m\left(t\right),t\in T^{\left(n\right)}\backslash o_m\right\}$, that is

$$S_{T^{\left(n\right)}\backslash o_m}\left(i_1,\dots ,i_m,i_{m+1}\right)=\sum _{k=1}^n\sum _{t\in L_k}{I}_{i_1}\left(X_{m_t}\right)\cdot \cdot \cdot I_{i_{m+1}}\left(X_t\right).$$

(53)

Corollary 3 Let {X _t , t ∈ T} be defined as before. Then for all i₁,..., i _{m +1} ∈ G, c ≥ 0, we have

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}|\frac{S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i_1,\dots ,i_m\right)}{\left|T^{\left(n\right)}\right|}-\frac{1}{\left|T^{\left(n-1\right)}\right|}\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in L_k}{I}_l\left(X_{{\left(m-1\right)}_t}\right)\\ .I_{i_1}\left(X_{{\left(m-2\right)}_t}\right)\cdot \cdot \cdot I_{i_{m-1}}\left(X_t\right)q_{k+1}\left(i_m|l,i_1,\dots ,i_{m-1}\right)|\le c+2\sqrt{c},P-a.e.,\omega \in D\left(c\right).\end{array}$$

(54)

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}|\frac{S_{T^{\left(n\right)}\backslash o_m}\left(i_1,\dots ,i_{m+1}\right)}{\left|T^{\left(n\right)}\right|}-\frac{1}{\left|T^{\left(n-1\right)}\right|}\sum _{k=0}^{n-1}\sum _{t\in L_k}{I}_{i_1}\left(X_{{\left(m-1\right)}_t}\right)\\ .I_{i_2}\left(X_{{\left(m-2\right)}_t}\right)\cdot \cdot \cdot I_{i_m}\left(X_t\right)q_{k+1}\left(i_{m+1}|i_1,\dots ,i_m\right)|\le c+2\sqrt{c},P-a.e.,\omega \in D\left(c\right).\end{array}$$

(55)

Proof Letting $g\left(y_1,\dots ,y_{m+1}\right)=I_{i_1}\left(y_2\right)\cdot \cdot \cdot I_{i_m}\left(y_{m+1}\right)$ in Corollary 2.

$$\begin{array}{ll}\hfill \sum _{k=1}^n\sum _{t\in L_k}g\left(X_0^m\left(t\right)\right)& =\sum _{k=1}^n\sum _{t\in L_k}{I}_{i_1}\left(X_{{\left(m-1\right)}_t}\right)\cdot \cdot \cdot I_{i_m}\left(X_t\right)\\ =S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i_1,\dots ,i_m\right)-I_{i_1}\left(X_{-\left(m-1\right)}\right)\cdot \cdot \cdot I_{i_m}\left(X_o\right),\end{array}$$

(56)

and

$$\begin{array}{c}\sum _{k=1}^n\sum _{t\in L_k}{E}_Q\left[g\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\\ =\sum _{k=1}^n\sum _{t\in L_k}\sum _{x_t\in G}g\left(X_1^m\left(t\right),x_t\right)q_k\left(x_t|X_1^m\left(t\right)\right)\\ =\sum _{k=1}^n\sum _{t\in L_k}\sum _{x_t\in G}{I}_{i_1}\left(X_{{\left(m-1\right)}_t}\right)\cdot \cdot \cdot I_{i_{m-1}}\left(X_{1_t}\right)I_{i_m}\left(x_t\right)q_k\left(x_t|X_1^m\left(t\right)\right)\\ =\sum _{k=1}^n\sum _{t\in L_k}{I}_{i_1}\left(X_{{\left(m-1\right)}_t}\right)\cdot \cdot \cdot I_{i_{m-1}}\left(X_{1_t}\right)q_k\left(i_m|X_1^m\left(t\right)\right)\\ =\sum _{l\in G}\sum _{k=1}^n\sum _{t\in L_k}{I}_l\left(X_{m_t}\right)I_{i_1}\left(X_{{\left(m-1\right)}_t}\right)\cdot \cdot \cdot I_{i_{m-1}}\left(X_{1_t}\right)q_k\left(i_m|l,i_1,\dots ,i_{m-1}\right)\\ =N\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in L_k}{I}_l\left(X_{{\left(m-1\right)}_t}\right)I_{i_1}\left(X_{{\left(m-2\right)}_t}\right)\cdot \cdot \cdot I_{i_{m-1}}\left(X_t\right)q_{k+1}\left(i_m|l,i_1,\dots ,i_{m-1}\right).\end{array}$$

(57)

Noticing that M = max g(y₁,..., y _{m +1}) = 1, $\underset{n\to \infty }{\text{lim}}\frac{\left|T^{\left(n-1\right)}\right|}{\left|T^{\left(n\right)}\right|}=\frac{1}{N}$, by (56) and (57) and Corollary 2, (54) holds. Similarly, we let $g\left(y_1,\dots ,y_{m+1}\right)=I_{i_1}\left(y_1\right)\cdot \cdot \cdot I_{i_{m+1}}\left(y_{m+1}\right),$ (55) follows.

Corollary 4 Let {X _t , t ∈ T} be defined as before.

$$\underset{n\to \infty }{\text{lim}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\left\{g\left(X_0^m\left(t\right)\right)-E_Q\left[g\left(X_0^m\left(t\right)\right)|X_1^m\left(t\right)\right]\right\}=0,P-a.e.,\omega \in D\left(0\right),$$

(58)

$$\begin{array}{l}\underset{n\to \infty }{\lim }\{\frac{S_{T^{(n)}\backslash {o^{\prime }}_m}(i_1\mathrm{,}\dots \mathrm{,}{i}_m)}{\left|T^{(n)}\right|}-\frac{1}{\left|T^{(n-1)}\right|}{\displaystyle \sum _{l\in G}{\displaystyle \sum _{k=0}^{n-1}{\displaystyle \sum _{t\in L_k}{I}_l(X_{{(m-1)}_t})}}}\\ .I_{i_1}(X_{{(m-2)}_t})\cdot \cdot \cdot I_{i_{m-1}}(X_t)q_{k+1}(i_m|l\mathrm{,}{i}_1\mathrm{,...,}{i}_{m-1})\}=0,P-a\mathrm{.}e\mathrm{.,}\omega \in D(0),\end{array}$$

(59)

$$\begin{array}{l}\underset{n\to \infty }{\lim }\{\frac{S_{T^{(n)}\backslash o_m}(i_1\mathrm{,}\dots \mathrm{,}{i}_{m+1})}{\left|T^{(n)}\right|}-\frac{1}{\left|T^{(n-1)}\right|}{\displaystyle \sum _{k=0}^{n-1}{\displaystyle \sum _{t\in L_k}{I}_{i_1}(X_{{(m-1)}_t})}}\\ .I_{i_2}(X_{{(m-2)}_t})\cdot \cdot \cdot I_{i_m}(X_t)q_{k+1}(i_{m+1}|i_1\mathrm{,}\dots \mathrm{,}{i}_m)\}=0,P-a\mathrm{.}e.,\omega \in D(0).\end{array}$$

(60)

If P = Q, then above equations hold Q - a.e..

Proof Letting c = 0 in Corollary 2 and Corollary 3, (58)-(60) follow from (51),(54) and (55). In particular, if P = Q, then h(P|Q) = 0, so (58)-(60) hold P - a.e., hence hold Q - a.e.

Definition 3 Let G = {0, 1,..., b - 1} be a finite state space and

$$Q_1=\left(q\left(j|i^m\right)\right),j\in G,i^m\in G^m$$

(61)

be an m th-order transition matrix. Define a stochastic matrix as follows:

$${\bar{Q}}_1=\left(q\left(j^m|i^m\right)\right),i^m,j^m\in G^m,$$

(62)

where

$$q\left(j^m|i^m\right)=\left\{\begin{array}{cc}\hfill q\left(j_m|i^m\right),\hfill & \hfill \mathsf{\text{if}}{j}_v=i_{v+1},v=1,2,\dots ,m-1,\hfill \\ \hfill 0,\hfill & \hfill \mathsf{\text{otherwise}}.\hfill \end{array}\right.$$

(63)

Then ${\bar{Q}}_1$ is called an m-dimensional stochastic matrix determined by the m th-order transition matrix.Q₁.

Lemma 2 (see [16]). Let ${\bar{Q}}_{\mathsf{\text{1}}}$ be an m-dimensional stochastic matrix determined by the m th-order transition matrix Q₁. If the elements of Q₁ are all positive, that is

$$Q_1=\left(q\left(j|i^m\right)\right),q\left(j|i^m\right)>0,\forall j\in G,i^m\in G^m,$$

(64)

then ${\bar{Q}}_1$ is ergodic.

Theorem 4 Let {X _t , t ∈ T} be defined as Theorem 1. Let $S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i_1,\dots ,i_m\right)={S_T}_{\left(n\right)\backslash {o^{\prime }}_m}\left(i^m\right),{S_T}_{\left(n\right)\backslash o_m}\left(i_1,\dots ,i_m,i_{m+1}\right)={S_T}_{\left(n\right)\backslash o_m}\left(i^{m+1}\right)$ and f _n (ω) defined by (52),(53) and (2), respectively. Let h(P|Q) and D(c) be defined by (4) and (12), respectively. Let the m th-order transition matrices defined by (6) be changeless with n, that is

$$Q_n=Q_1=\left(q\left(j|i^m\right)\right),$$

(65)

or {X _t , t ∈ T} is an m th-order homogeneous Markov chain indexed by tree T with the m th-order transition matrix Q₁ under the probability measure Q. Let the m-dimensional stochastic matrix ${\bar{Q}}_1$ determined by Q₁ be ergodic. Then for all i₁,..., i _{m +1} ∈ G, we have

$${\text{lim}}_{n\to \infty }\frac{S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i^m\right)}{\left|T^{\left(n\right)}\right|}=\pi \left(i^m\right),P-a.e.,\omega \in D\left(0\right).$$

(66)

$$\underset{n\to \infty }{\text{lim}}\frac{{S_T}_{\left(n\right)\backslash o_m}\left(i^{m+1}\right)}{\left|T^{\left(n\right)}\right|}=\pi \left(i^m\right)q\left(i_{m+1}|i^m\right),P-a.e.,\omega \in D\left(0\right).$$

(67)

$$\underset{n\to \infty }{\text{lim}}{f}_n\left(\omega \right)=-\sum _{i^m\in G^m}\sum _{j\in G}\pi \left(i^m\right)q\left(j|i^m\right)\text{ln}q\left(j|i^m\right),P-a.e.,\omega \in D\left(0\right).$$

(68)

where {π(i^m), i^m ∈ G^m} is the stationary distribution determined by ${\bar{Q}}_1$.

Proof Proof of Equation 66. Let k^m = (k₁,..., k _m ). If (65) holds, then we have by (63) and (52)

$$\begin{array}{c}\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in L_k}{I}_l\left(X_{{\left(m-1\right)}_t}\right)I_{i_1}\left(X_{{\left(m-2\right)}_t}\right)\cdot \cdot \cdot I_{i_{m-1}}\left(X_t\right)q_{k+1}\left(i_m|l,i_1,\dots ,i_{m-1}\right)\\ =\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in L_k}{I}_l\left(X_{{\left(m-1\right)}_t}\right)I_{i_1}\left(X_{{\left(m-2\right)}_t}\right)\cdot \cdot \cdot I_{i_{m-1}}\left(X_t\right)q\left(i_m|l,i_1,\dots ,i_{m-1}\right)\\ =\sum _{l\in G}{S}_{T^{\left(n-1\right)}\backslash {o^{\prime }}_m}\left(l,i_1,\cdots ,i_{m-1}\right)q\left(i_m|l,i_1,\dots ,i_{m-1}\right)\\ =\sum _{k^m\in G^m}{S}_{T^{\left(n-1\right)}\backslash {o^{\prime }}_m}\left(k^m\right)q\left(i^m|k^m\right).\end{array}$$

(69)

By (59) and (69), we have

$$\underset{n\to \infty }{\text{lim}}\left\{\frac{S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i^m\right)}{\left|T^{\left(n\right)}\right|}-\frac{1}{\left|T^{\left(n-1\right)}\right|}\sum _{k^m\in G^m}{S}_{T\left(n-1\right)\backslash {o^{\prime }}_m}\left(k^m\right)q\left(i^m|k^m\right)\right\}=0,P-a.e.,\omega \in D\left(0\right).$$

(70)

Multiplying (70) by q(j^m|i^m), adding them together for i^m ∈ G^m, and using (70) once again, we have

$$\begin{array}{ll}\hfill 0& =\sum _{i^m\in G^m}q\left(j^m|i^m\right)\cdot \underset{n\to \infty }{\text{lim}}\left\{\frac{S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i^m\right)}{\left|T^{\left(n\right)}\right|}-\frac{1}{\left|T^{\left(n-1\right)}\right|}\sum _{k^m\in G^m}{S}_{T^{\left(n-1\right)}\backslash {o^{\prime }}_m}\left(k^m\right)q\left(i^m|k^m\right)\right\}\\ =\underset{n\to \infty }{\text{lim}}\left\{\sum _{i^m\in G^m}\frac{S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i^m\right)}{\left|T^{\left(n\right)}\right|}q\left(j^m|i^m\right)-\frac{S_{T^{\left(n+1\right)}\backslash {o^{\prime }}_m}\left(j^m\right)}{\left|T^{\left(n+1\right)}\right|}\right\}\\ +\underset{n\to \infty }{\text{lim}}\left\{\frac{S_{T^{\left(n+1\right)}\backslash {o^{\prime }}_m}\left(j^m\right)}{\left|T^{\left(n+1\right)}\right|}-\frac{1}{\left|T^{\left(n-1\right)}\right|}\sum _{k^m\in G^m}{S}_{T^{\left(n-1\right)}\backslash {o^{\prime }}_m}\left(k^m\right)\sum _{i^m\in G^m}q\left(j^m|i^m\right)q\left(i^m|k^m\right)\right\}\\ =\underset{n\to \infty }{\text{lim}}\left\{\frac{S_{T^{\left(n+1\right)}\backslash {o^{\prime }}_m}\left(j^m\right)}{\left|T^{\left(n+1\right)}\right|}-\frac{1}{\left|T^{\left(n-1\right)}\right|}\sum _{k^m\in G^m}{S}_{T^{\left(n-1\right)}\backslash {o^{\prime }}_m}\left(k^m\right)q^{\left(2\right)}\left(j^m|k^m\right)\right\},P-a.e.,\omega \in D\left(0\right).\end{array}$$

By induction, we have

$$\underset{n\to \infty }{\text{lim}}\left\{\frac{S_{T^{\left(n+N\right)}\backslash {o^{\prime }}_m}\left(j^m\right)}{\left|T^{\left(n+N\right)}\right|}-\frac{1}{\left|T^{\left(n-1\right)}\right|}\sum _{k^m\in G^m}{S}_{T^{\left(n-1\right)}\backslash {o^{\prime }}_m}\left(k^m\right)q^{\left(N+1\right)}\left(j^m|k^m\right)\right\}=0,P-a.e.,\omega \in D\left(0\right).$$

(71)

where q^(h)(j^m|k^m) is the h th step probability determined by ${\bar{Q}}_1$. We have by ergodicity

$$\underset{N\to \infty }{\text{lim}}{q}^{\left(N+1\right)}\left(j^m|k^m\right)=\pi \left(j^m\right),\forall k^m\in G^m,$$

(72)

and ${\sum }_{k^m\in G^m}{S}_{T^{\left(n-1\right)}\backslash {o^{\prime }}_m}\left(k^m\right)=\left|T^{\left(n-1\right)}\right|-\left(m-1\right)$. (66) follows from (71) and (72). By (66) and (60), Equation 67 follows easily.

Proof of Equation 68. By (66) and (53), we have

$$\begin{array}{l}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}H[q_k(0|X_1^m(t)),\dots \mathrm{,}{q}_k(b-1|X_1^m(t)))]}}\\ ={\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}H[q(\text{0}|X_1^m(t)),\dots \mathrm{,}q(b-1|X_1^m(t)))]}}\\ =-{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}{\displaystyle \sum _{j\in G}q(j|X_1^m(t))\ln q(j|X_1^m(t))}}}\\ =-{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{t\in L_k}{\displaystyle \sum _{j\in G}{\displaystyle \sum _{i^m\in G^m}{I}_{i_1}(X_{m_t})\cdot \cdot \cdot I_{i_m}(X_{1_t})q(j|i^m)\ln q(j|i^m)}}}}\\ =-N{\displaystyle \sum _{k=0}^{n–1}{\displaystyle \sum _{t\in L_k}{\displaystyle \sum _{j\in G}{\displaystyle \sum _{i^m\in G^m}{I}_{i_1}(X_{{(m-1)}_t})\cdots I_{i_m}(X_t)q(j|i^m)\ln q(j|i^m)}}}}\\ =-N{\displaystyle \sum _{j\in G}{\displaystyle \sum _{i^m\in G^m}{S}_T(n-1)\backslash {o\prime }_m(i^m)q(j|i^m)\ln q(j|i^m).}}\end{array}$$

(73)

Noticing that $\underset{n\to \infty }{\text{lim}}\frac{\left|T^{\left(n-1\right)}\right|}{\left|T^{\left(n\right)}\right|}=\frac{1}{N}$, by (39), (73) and (66), Equation 68 follows.□

3. Shannon-McMillan Theorem

Theorem 5 Let {X _t , t ∈ T} be a G-valued m th-order nonhomogeneous Markov chain indexed by an m rooted Cayley tree under the probability measure Q with initial distribution (5) and m th-order transition matrices (6). Let $S_{T^{\left(n\right)}}\backslash o_m^{\prime }\left(i^m\right),{S_T}_{\left(n\right)}\backslash o_m\left(i^{m+1}\right)$ and f _n (ω) be defined as before. Let

$$Q_n=Q_1=\left(q\left(j|i^m\right)\right),q\left(j|i^m\right)>0,\forall i^m\in G^m,j\in G,$$

(74)

be another positive m th-order transition matrix. Let ${\bar{Q}}_1$ be an m dimension transition matrix determined by Q₁. If

$$\underset{n\to \infty }{\text{lim}}{q}_n\left(j|i^m\right)=q\left(j|i^m\right),\forall i^m\in G^m,j\in G,$$

(75)

then

$$\underset{n\to \infty }{\text{lim}}\frac{S_{T^{\left(n\right)}\backslash {o^{\prime }}_m}\left(i^m\right)}{\left|T^{\left(n\right)}\right|}=\pi \left(i^m\right),P-a.e.\omega \in D\left(0\right),$$

(76)

$$\underset{n\to \infty }{\text{lim}}\frac{S_{T^{\left(n\right)}\backslash o_m}\left(i^{m+1}\right)}{\left|T^{\left(n\right)}\right|}=\pi \left(i^m\right)q\left(i_{m+1}|i^m\right),P-a.e.\omega \in D\left(0\right),$$

(77)

$$\underset{n\to \infty }{\text{lim}}{f}_n\left(\omega \right)=-\sum _{i^m\in G^m}\sum _{j\in G}q\left(j|i^m\right)\text{ln}q\left(j|i^m\right),P-a.e.\omega \in D\left(0\right),$$

(78)

where {π(i^m), i^m ∈ G^m} is the stationary distribution determined by ${\bar{Q}}_1$. In particular, if P = Q, then above equations hold Q - a.e.

Proof By (59), (75), (52) and (66), (76) follows immediately. Similarly, by (60), (75), and (53), (77) follows. It follows from (75) and Cesaro average that

$$\underset{n\to \infty }{\text{lim}}\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}|q_k\left(j|i^m\right)\text{ln}{q}_k\left(j|i^m\right)-q_k\left(j|i^m\right)\text{ln}{q}_k\left(j|i^m\right)|=0,\forall i^m\in G^m,j\in G.$$

(79)

Notice that

$$\begin{array}{c}|\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q_k\left(0|X_1^m\left(t\right)\right),\dots ,q_k\left(0|X_1^m\left(t\right)\right)\right]-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}H\left[q\left(0|X_1^m\left(t\right)\right),\dots ,q\left(0|X_1^m\left(t\right)\right)\right]|\\ =|-\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\sum _{j\in G}{q}_k\left(j|X_1^m\left(t\right)\right)\text{ln}{q}_k\left(j|X_1^m\left(t\right)\right)+\frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\sum _{j\in G}q\left(j|X_1^m\left(t\right)\right)\text{ln}q\left(j|X_1^m\left(t\right)\right)\\ \le \frac{1}{\left|T^{\left(n\right)}\right|}\sum _{k=1}^n\sum _{t\in L_k}\sum _{j\in G}\sum _{i^m\in G^m}|q_k\left(j|i^m\right)\text{ln}{q}_k\left(j|i^m\right)-q\left(j|i^m\right)\text{ln}q\left(j|i^m\right)|.\end{array}$$

(80)

By (79), (80), (39), (73) and (66), (78) follows. In particular, if P = Q, then h(P|Q) = 0. (76), (77) and (78) also holds P - a.e. □