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

Zhiyan Shi¹Email author,
Weiguo Yang¹,
Lixin Tian¹ and
Weicai Peng²

Journal of Inequalities and Applications20122012:1

https://doi.org/10.1186/1029-242X-2012-1

Received: 21 March 2011
Accepted: 4 January 2012
Published: 4 January 2012

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.

Keywords

strong deviation theorem
m rooted Cayley tree
m th-order nonhomogeneous Markov chain
Shannon-McMillan theorem

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 $\bar{σ t}$ 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 $\bar{σ t}$ .

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

{\bar{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

(Ω, F)

be a measure space, {X_t, t∈T} be a collection of random variables defined on

(Ω, F)

and taking values in G = {0,1,..., b - 1}, where b is a positive integer. Let P be a general probability distribution on

(Ω, F)

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

(1)

Let

f_{n} (ω) = - \frac{1}{| T^{(n)} |} ln P (X^{T^{(n)}}) .

(2)

f_n(ω) is called entropy density of $X^{T^{(n)}}$ .

Let Q be another probability measure on the measurable space

(Ω, F)

, and let the distribution of {X_t, t ∈ T} under Q be

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

(3)

Let

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

(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 = (P (y | x_{1}, x_{2}, \dots, x_{m})), P (y | x_{1}, x_{2}, \dots, x_{m}) \geq 0, x_{1}, x_{2}, \dots, x_{m}, y \in G .

\sum_{y \in G} P (y | x_{1}, x_{2}, \dots, x_{m}) = 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 $(Ω, F, Q)$ . Let Q be a probability on a measurable space $(Ω, F)$ .

Let

q = (q (x_{1}, x_{2}, \dots, x_{m})), x_{1}, x_{2}, \dots, x_{m} \in G

(5)

be a distribution on G^m, and

Q_{n} = (q_{n} (y | x_{1}, x_{2}, \dots, x_{m})), x_{1}, x_{2}, \dots, x_{m}, y \in G, n \geq 1

(6)

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

\begin{gathered} Q (X_{t} = y | X_{1_{t}} = x_{1}, X_{2_{t}} = x_{2}, \dots, X_{m_{t}} = x_{m} and X_{σ} for σ \land t \leq 1_{t}) \\ = Q (X_{t} = y | X_{1_{t}} = x_{1}, X_{2_{t}} = x_{2}, \dots, X_{m_{t}} = x_{m}) \\ = q_{n} (y | x_{1}, x_{2}, \dots, x_{m}), \forall x_{1}, x_{2}, \dots, x_{m}, y \in G \end{gathered}

(7)

and

\begin{gathered} Q (X_{- (m - 1)} = x_{1}, \dots, X_{- 1} = x_{m - 1}, X_{o} = x_{m}) \\ = q (x_{1}, \dots, x_{m - 1}, x_{m}), x_{1}, \dots, x_{m} \in G, \end{gathered}

(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} = {o, - 1, - 2, \dots, - (m - 1)}, o_{m}^{'} = {- 1, - 2, \dots, - (m - 1)},

X_{1}^{n} (t) = {X_{n_{t}}, \dots, X_{2_{t}}, X_{1_{t}}}, X_{0}^{n} (t) = {X_{n_{t}}, \cdot \cdot \cdot, X_{2_{t}}, X_{1_{t}}, X_{t}},

and denote by $x_{1}^{n} (t)$ and $x_{0}^{n} (t)$ the realizations $X_{1}^{n} (t)$ and $X_{0}^{n} (t)$ , 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 (x^{T^{(n)}}) = Q (X^{T^{(n)}} = x^{T^{(n)}}) = q (x_{- (m - 1)}, \dots, x_{o}) \prod_{k = 1}^{n} \prod_{t \in L_{k}} q_{k} (x_{t} | x_{1}^{m} (t)) .

(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

(Ω, F)

. Let P and Q be two probability measures on the measurable space

(Ω, F)

, 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} = σ (X^{T^{(n)}}) (n \geq 1)

. Set

F_{n} (ω) = \sum_{k = 1}^{n} \sum_{t \in L_{k}} g_{k} (X_{0}^{m} (t))

(10)

and

t_{n} (λ, ω) = \frac{e^{λ F_{n} (ω)}}{\prod_{k = 1}^{n} \prod_{t \in L_{k}} E_{Q} [e^{λ g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t)]} \cdot \frac{q (X_{- (m - 1)}, \dots, X_{o}) \prod_{k = 1}^{n} \prod_{t \in L_{k}} q_{k} (X_{t} | X_{1}^{m} (t))}{P (x^{T^{(n)}})},

(11)

where E_Q denote the expectation under probability measure Q. Then ${t_{n} (λ, ω), F_{n}, n \geq 1}$ 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

(Ω, F)

. Let P and Q be two probability measures on the measurable space

(Ω, F)

, 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 (c) = {ω : h (P | Q) \leq c} .

(12)

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

b_{α} (i^{m}) = \underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} E_{Q} [e^{a | g_{k} (X_{0}^{m} (t)) |} | X_{1}^{m} (t) = i^{m}] \leq τ .

(13)

Let

A_{t} = \frac{2 τ}{e^{2} {(t - α)}^{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}{| T^{(n)} |} | \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} | \lim \sup} \leq 2 \sqrt{c A_{t}}, P - a . e ., ω \in D (c) .

(15)

In particular,

lim_{n \to \infty} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} = 0, P - a . e ., ω \in D (0) .

(16)

Proof Let t_n(λ, ω) be defined by (11). By Lemma 1,

{t_{n} (λ, ω), F_{n}, n \geq 1}

is a non-negative martingale under probability measure P. By Doob's martingale convergence theorem, we have

lim_{n \to \infty} t_{n} (λ, ω) = t (λ, ω) < \infty, P - a . e .

Hence,

\underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} ln t_{n} (λ, ω) \leq 0, P - a . e . .

(17)

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

\underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} [\sum_{k = 1}^{n} \sum_{t \in L_{k}} \{λ g_{k} (X_{0}^{m} (t)) - ln E_{Q} [e^{λ g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t)]\} - ln \frac{P (X^{T^{(n)}})}{Q (X^{T (n)})}] \leq 0, P - a . e .

(18)

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

\underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} \{λ g_{k} (X_{0}^{m} (t)) - ln E_{Q} [e^{λ g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t)]\} \leq c, P - a . e ., ω \in D (c) .

(19)

This implies that

\begin{gathered} \underset{n \to \infty}{lim sup} \frac{λ}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq \underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} \{ln E_{Q} [e^{λ g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t)] - E_{Q} [λ g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]\} + c, P - a . e ., ω \in D (c) \end{gathered}

(20)

Let |λ| < t. By inequalities In x ≤ x -1(x > 0) and

e^{x} - 1 - x \leq \frac{x^{2}}{2} e^{| x |}

, and noticing that

max {x^{2} e^{- h x}, x \geq 0} = 4 e^{- 2} / h^{2} (h > 0) .

(21)

We have

\begin{array}{l} \underset{n \to \infty}{\lim \sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {\ln E_{Q} [e^{λ g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t)] - E_{Q} [λ g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq \underset{n \to \infty}{\lim \sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {E_{Q} [e^{λ g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t)] - 1 - E_{Q} [λ g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq \frac{λ^{2}}{2} \underset{n \to \infty}{\lim \sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} E_{Q} [g_{k} 2 (X_{0}^{m} (t)) e^{| λ {| |}_{g_{k}} (X_{0}^{m} (t)) |} | X_{1}^{m} (t)] \\ = \frac{λ^{2}}{2} \underset{n \to \infty}{\lim \sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} E_{Q} [e^{α | g_{k} (X_{0}^{m} (t)) |} g_{k} 2 (X_{0}^{m} (t)) e^{(| λ | - α) | g_{k} (X_{0}^{m} (t)) |} | X_{1}^{m} (t)] \\ \leq \frac{λ^{2}}{2} \underset{n \to \infty}{\lim \sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} E_{Q} [e^{α | g_{k} (X_{0}^{m} (t)) |} 4 e^{- 2} / {(| λ | - a)}^{2} | X_{1}^{m} (t)] \\ \leq 2 λ^{2} τ / e^{2} {(t - α)}^{2} . \end{array}

(22)

By (20) and (22), we have

\begin{gathered} \underset{n \to \infty}{lim sup} \frac{λ}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq λ^{2} A_{t} + c, P - a . e ., ω \in D (c) . \end{gathered}

(23)

When 0 < λ < t < α, we have by (23)

\begin{gathered} \underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq λ A_{t} + c / λ, P - a . e ., ω \in D (c) . \end{gathered}

(24)

It is easy to see that when 0 < c < t²A_t, the function f (λ) = λA_t + c/λ attains, at

λ = \sqrt{c / A_{t}}

, its smallest value

f (\sqrt{c / A_{t}}) = 2 \sqrt{c A_{t}}

. Letting

λ = \sqrt{c / A_{t}}

in (24), we have

\underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \leq 2 \sqrt{c A_{t}}, P - a . e ., ω \in D (c) .

(25)

When c = 0, we have by (24)

\underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \leq λ A_{t}, P - a . e ., ω \in D (0) .

(26)

Letting λ → 0⁺ in (26), we obtain

\underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \leq 0, P - a . e ., ω \in D (0) .

(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}{lim inf} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \geq - 2 \sqrt{c A_{t}}, P - a . e ., ω \in D (c) .

(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} = 2 b / e^{2} {(t - 1)}^{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{gathered} \underset{n \to \infty}{lim sup} {f_{n} (ω) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]} \\ \leq 2 \sqrt{c H_{t}}, P - a . e ., ω \in D (c), \end{gathered}

(30)

\begin{gathered} \underset{n \to \infty}{lim inf} {f_{n} (ω) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]} \\ \geq - 2 \sqrt{c H_{t}} - c, P - a . e ., ω \in D (c), \end{gathered}

(31)

where H(p₀,.... p_b_-1) denote the entropy of distribution (p₀,..., p_b_-1), i.e.,

H (p_{0}, \dots, p_{b - 1}) = - \sum_{i = 0}^{b - 1} p_{i} 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{gathered} E_{Q} [e^{g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t) = i^{m}] \\ = \sum_{j \in G} e^{| - ln q_{k} (j | i^{m}) |} q_{k} (j | i^{m}) \\ = \sum_{j \in G} q_{k} (j | i^{m}) / q_{k} (j | i^{m}) \\ = b . \end{gathered}

(32)

Hence, ∀i^m ∈ G^m,

b_{1} (i^{m}) = \underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} E_{Q} [e^{g_{k} (X_{0}^{m} (t))} | X_{1}^{m} (t) = i^{m}] \leq b .

(33)

Noticing that

\begin{gathered} E_{Q} [- ln q_{k} (X_{t} | X_{1}^{m} (t)) | X_{1}^{m} (t)] \\ = - \sum_{j \in G} q_{k} (j | X_{1}^{m} (t)) ln q_{k} (j | X_{1}^{m} (t)) \\ = H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))] . \end{gathered}

(34)

When 0 ≤ c ≤ t²H_t, we have by (34),(29) and (15)

\begin{gathered} \underset{n \to \infty}{lim sup} \{\frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} (- ln q_{k} (X_{t} | X_{1}^{m} (t))) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]\} \\ \leq 2 \sqrt{c H_{t}}, P - a . e ., ω \in D (c) . \end{gathered}

(35)

\begin{gathered} \underset{n \to \infty}{lim inf} \{\frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} (- ln q_{k} (X_{t} | X_{1}^{m} (t))) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]\} \\ \geq - 2 \sqrt{c H_{t}}, P - a . e ., ω \in D (c) . \end{gathered}

(36)

By (35), (9) and h(P|Q) ≥ 0,

\begin{array}{l} \underset{n \to \infty}{\lim \sup} {f_{n} (ω) - \frac{1}{| T^{(n)} |} |_{k = 1}^{n} \underset{t \in L_{k}}{|} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))] \\ \leq \underset{n \to \infty}{\lim \sup} {- \frac{1}{| T^{(n)} |} \ln P (X^{T^{(n)}}) - \frac{1}{| T^{(n)} |} |_{k = 1}^{n} \underset{t \in L_{k}}{|} (- \ln q_{k} (X_{t} | X_{1}^{m} (t))} \\ + \underset{n \to \infty}{\lim \sup} {\frac{1}{| T^{(n)} |} |_{k = 1}^{n} \underset{t \in L_{k}}{|} (- \ln q_{k} (X_{t} | X_{1}^{m} (t)) \\ - \frac{1}{| T^{(n)} |} |_{k = 1}^{n} \underset{t \in L_{k}}{|} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]} \\ \leq 2 \sqrt{c H_{t}}, P - a . e ., ω \in D (c) . \end{array}

(37)

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

\begin{array}{l} \underset{n \to \infty}{\lim \sup} {f_{n} (ω) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))] \\ \leq \underset{n \to \infty}{\lim \sup} {- \frac{1}{| T^{(n)} |} \ln P (X^{T^{(n)}}) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} (- \ln q_{k} (X_{t} | X_{1}^{m} (t))} \\ + \underset{n \to \infty}{\lim \sup} {\frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} (- \ln q_{k} (X_{t} | X_{1}^{m} (t)) \\ - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]} \\ \leq 2 \sqrt{c H_{t}}, P - a . e ., ω \in D (c) . \end{array}

(38)

This completes the proof of this theorem. □

Corollary 1 Under the conditions of Theorem 2, we have

lim_{n \to \infty} \{f_{n} (ω) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]\} = 0, P - a . e ., ω \in D (0) .

(39)

If P << Q, then

lim_{n \to \infty} \{f_{n} (ω) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]\} = 0, P - a . e .

(40)

In particular, if P = Q,

lim_{n \to \infty} \{f_{n} (ω) - \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t))]\} = 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}{lim sup} \frac{1}{| T^{(n)} |} | \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} | \leq M (c + 2 \sqrt{c}), P - a . e ., ω \in D (c) .

(42)

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

\begin{gathered} \underset{n \to \infty}{lim sup} \frac{λ}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq \underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} E_{Q} [e^{λ g_{k} (X_{0}^{m} (t))} - 1 - λ g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)] \\ + c P - a . e ., ω \in D (c) . \end{gathered}

(43)

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

e^{λ g_{k} (X_{0}^{m} (t))} - 1 - λ g_{k} (X_{0}^{m} (t)) \leq | λ | M (e^{| λ | M} - 1) .

(44)

By (43) and (44)

\begin{gathered} \underset{n \to \infty}{lim sup} \frac{λ}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq | λ | M (e^{| λ | | M} - 1) + c, P - a . e ., ω \in D (c) . \end{gathered}

(45)

When λ > 0, we have by (45)

\begin{gathered} \underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq M (e^{λ M} - 1) + c / λ, P - a . e ., ω \in D (c) . \end{gathered}

(46)

Taking

λ = \frac{1}{M} log (1 + \sqrt{c})

, and using the inequality

log (1 + \sqrt{c}) \geq \frac{\sqrt{c}}{1 + \sqrt{c}},

(47)

we have when c > 0

\begin{gathered} \underset{n \to \infty}{lim sup} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \leq M \sqrt{c} + \frac{c M}{log (1 + \sqrt{c})} \\ \leq M (2 \sqrt{c} + c), P - a . e ., ω \in D (c) . \end{gathered}

(48)

When λ < 0, it follows from (45) that

\begin{gathered} \underset{n \to \infty}{lim inf} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \geq - M (e^{λ M} - 1) + c / λ P - a . e ., ω \in D (c) . \end{gathered}

(49)

Taking

λ = - \frac{1}{M} log (1 + \sqrt{c})

in (49), and using (47), we have when c > 0

\begin{gathered} \underset{n \to \infty}{lim inf} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g_{k} (X_{0}^{m} (t)) - E_{Q} [g_{k} (X_{0}^{m} (t)) | X_{1}^{m} (t)]} \\ \geq - M \sqrt{c} - \frac{c M}{log (1 + \sqrt{c})} \\ \geq - M (2 \sqrt{c} + c), P - a . e ., ω \in D (c) . \end{gathered}

(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}{lim sup} \frac{1}{| T^{(n)} |} |\sum_{k = 1}^{n} \sum_{t \in L_{k}} {g (X_{0}^{m} (t)) - E_{Q} [g (X_{0}^{m} (t)) | X_{1}^{m} (t)]}| \leq M (c + 2 \sqrt{c}), P - a . e ., ω \in D (c) .

(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} (x) = \{\begin{matrix} 1 & x = k \\ 0 & x \neq k \end{matrix}

. Let

S_{T^{(n)} \ {o^{'}}_{m}} (i_{1}, \dots, i_{m})

be the number of (i₁,..., i_m) in the collection of

{X_{0}^{m - 1} (t), t \in T^{(n)} \ o_{m}^{'}}

, that is

S_{T^{(n)} \ {o^{'}}_{m}} (i_{1}, \dots, i_{m}) = \sum_{k = 0}^{n} \sum_{t \in L_{k}} I_{i_{1}} (X_{{(m - 1)}_{t}}) \cdot \cdot \cdot I_{i_{m}} (X_{t}),

(52)

S_{T^{(n)} \ o_{m}} (i_{1}, \dots, i_{m}, i_{m + 1})

be the number of (i₁,..., i_m, i_m₊₁) in the collection of

{X_{0}^{m} (t), t \in T^{(n)} \ o_{m}}

, that is

S_{T^{(n)} \ o_{m}} (i_{1}, \dots, i_{m}, i_{m + 1}) = \sum_{k = 1}^{n} \sum_{t \in L_{k}} I_{i_{1}} (X_{m_{t}}) \cdot \cdot \cdot I_{i_{m + 1}} (X_{t}) .

(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{gathered} \underset{n \to \infty}{lim sup} | \frac{S_{T^{(n)} \ {o^{'}}_{m}} (i_{1}, \dots, i_{m})}{| T^{(n)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{l \in G} \sum_{k = 0}^{n - 1} \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, i_{1}, \dots, i_{m - 1}) | \leq c + 2 \sqrt{c}, P - a . e ., ω \in D (c) . \end{gathered}

(54)

\begin{gathered} \underset{n \to \infty}{lim sup} | \frac{S_{T^{(n)} \ o_{m}} (i_{1}, \dots, i_{m + 1})}{| T^{(n)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{k = 0}^{n - 1} \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}, \dots, i_{m}) | \leq c + 2 \sqrt{c}, P - a . e ., ω \in D (c) . \end{gathered}

(55)

Proof Letting

g (y_{1}, \dots, y_{m + 1}) = I_{i_{1}} (y_{2}) \cdot \cdot \cdot I_{i_{m}} (y_{m + 1})

in Corollary 2.

\begin{array}{l} \sum_{k = 1}^{n} \sum_{t \in L_{k}} g (X_{0}^{m} (t)) & = \sum_{k = 1}^{n} \sum_{t \in L_{k}} I_{i_{1}} (X_{{(m - 1)}_{t}}) \cdot \cdot \cdot I_{i_{m}} (X_{t}) \\ = S_{T^{(n)} \ {o^{'}}_{m}} (i_{1}, \dots, i_{m}) - I_{i_{1}} (X_{- (m - 1)}) \cdot \cdot \cdot I_{i_{m}} (X_{o}), \end{array}

(56)

and

\begin{gathered} \sum_{k = 1}^{n} \sum_{t \in L_{k}} E_{Q} [g (X_{0}^{m} (t)) | X_{1}^{m} (t)] \\ = \sum_{k = 1}^{n} \sum_{t \in L_{k}} \sum_{x_{t} \in G} g (X_{1}^{m} (t), x_{t}) q_{k} (x_{t} | X_{1}^{m} (t)) \\ = \sum_{k = 1}^{n} \sum_{t \in L_{k}} \sum_{x_{t} \in G} I_{i_{1}} (X_{{(m - 1)}_{t}}) \cdot \cdot \cdot I_{i_{m - 1}} (X_{1_{t}}) I_{i_{m}} (x_{t}) q_{k} (x_{t} | X_{1}^{m} (t)) \\ = \sum_{k = 1}^{n} \sum_{t \in L_{k}} I_{i_{1}} (X_{{(m - 1)}_{t}}) \cdot \cdot \cdot I_{i_{m - 1}} (X_{1_{t}}) q_{k} (i_{m} | X_{1}^{m} (t)) \\ = \sum_{l \in G} \sum_{k = 1}^{n} \sum_{t \in L_{k}} I_{l} (X_{m_{t}}) I_{i_{1}} (X_{{(m - 1)}_{t}}) \cdot \cdot \cdot I_{i_{m - 1}} (X_{1_{t}}) q_{k} (i_{m} | l, i_{1}, \dots, i_{m - 1}) \\ = N \sum_{l \in G} \sum_{k = 0}^{n - 1} \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, i_{1}, \dots, i_{m - 1}) . \end{gathered}

(57)

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

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

lim_{n \to \infty} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} {g (X_{0}^{m} (t)) - E_{Q} [g (X_{0}^{m} (t)) | X_{1}^{m} (t)]} = 0, P - a . e ., ω \in D (0),

(58)

\begin{array}{l} \lim_{n \to \infty} {\frac{S_{T^{(n)} \ {o^{'}}_{m}} (i_{1}, \dots, i_{m})}{| T^{(n)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{l \in G} \sum_{k = 0}^{n - 1} \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, i_{1},..., i_{m - 1})} = 0, P - a . e ., ω \in D (0), \end{array}

(59)

\begin{array}{l} \lim_{n \to \infty} {\frac{S_{T^{(n)} \ o_{m}} (i_{1}, \dots, i_{m + 1})}{| T^{(n)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{k = 0}^{n - 1} \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}, \dots, i_{m})} = 0, P - a . e ., ω \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} = (q (j | i^{m})), 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} = (q (j^{m} | i^{m})), i^{m}, j^{m} \in G^{m},

(62)

where

q (j^{m} | i^{m}) = \{\begin{matrix} q (j_{m} | i^{m}), & if j_{v} = i_{v + 1}, v = 1, 2, \dots, m - 1, \\ 0, & otherwise . \end{matrix}

(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}}_{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} = (q (j | i^{m})), q (j | i^{m}) > 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^{(n)} \ {o^{'}}_{m}} (i_{1}, \dots, i_{m}) = {S_{T}}_{(n) \ {o^{'}}_{m}} (i^{m}), {S_{T}}_{(n) \ o_{m}} (i_{1}, \dots, i_{m}, i_{m + 1}) = {S_{T}}_{(n) \ o_{m}} (i^{m + 1})

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} = (q (j | i^{m})),

(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

{lim}_{n \to \infty} \frac{S_{T^{(n)} \ {o^{'}}_{m}} (i^{m})}{| T^{(n)} |} = π (i^{m}), P - a . e ., ω \in D (0) .

(66)

lim_{n \to \infty} \frac{{S_{T}}_{(n) \ o_{m}} (i^{m + 1})}{| T^{(n)} |} = π (i^{m}) q (i_{m + 1} | i^{m}), P - a . e ., ω \in D (0) .

(67)

lim_{n \to \infty} f_{n} (ω) = - \sum_{i^{m} \in G^{m}} \sum_{j \in G} π (i^{m}) q (j | i^{m}) ln q (j | i^{m}), P - a . e ., ω \in D (0) .

(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{gathered} \sum_{l \in G} \sum_{k = 0}^{n - 1} \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, i_{1}, \dots, i_{m - 1}) \\ = \sum_{l \in G} \sum_{k = 0}^{n - 1} \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 (i_{m} | l, i_{1}, \dots, i_{m - 1}) \\ = \sum_{l \in G} S_{T^{(n - 1)} \ {o^{'}}_{m}} (l, i_{1}, \dots, i_{m - 1}) q (i_{m} | l, i_{1}, \dots, i_{m - 1}) \\ = \sum_{k^{m} \in G^{m}} S_{T^{(n - 1)} \ {o^{'}}_{m}} (k^{m}) q (i^{m} | k^{m}) . \end{gathered}

(69)

By (59) and (69), we have

lim_{n \to \infty} \{\frac{S_{T^{(n)} \ {o^{'}}_{m}} (i^{m})}{| T^{(n)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{k^{m} \in G^{m}} S_{T (n - 1) \ {o^{'}}_{m}} (k^{m}) q (i^{m} | k^{m})\} = 0, P - a . e ., ω \in D (0) .

(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}{l} 0 & = \sum_{i^{m} \in G^{m}} q (j^{m} | i^{m}) \cdot lim_{n \to \infty} \{\frac{S_{T^{(n)} \ {o^{'}}_{m}} (i^{m})}{| T^{(n)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{k^{m} \in G^{m}} S_{T^{(n - 1)} \ {o^{'}}_{m}} (k^{m}) q (i^{m} | k^{m})\} \\ = lim_{n \to \infty} \{\sum_{i^{m} \in G^{m}} \frac{S_{T^{(n)} \ {o^{'}}_{m}} (i^{m})}{| T^{(n)} |} q (j^{m} | i^{m}) - \frac{S_{T^{(n + 1)} \ {o^{'}}_{m}} (j^{m})}{| T^{(n + 1)} |}\} \\ + lim_{n \to \infty} \{\frac{S_{T^{(n + 1)} \ {o^{'}}_{m}} (j^{m})}{| T^{(n + 1)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{k^{m} \in G^{m}} S_{T^{(n - 1)} \ {o^{'}}_{m}} (k^{m}) \sum_{i^{m} \in G^{m}} q (j^{m} | i^{m}) q (i^{m} | k^{m})\} \\ = lim_{n \to \infty} \{\frac{S_{T^{(n + 1)} \ {o^{'}}_{m}} (j^{m})}{| T^{(n + 1)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{k^{m} \in G^{m}} S_{T^{(n - 1)} \ {o^{'}}_{m}} (k^{m}) q^{(2)} (j^{m} | k^{m})\}, P - a . e ., ω \in D (0) . \end{array}

By induction, we have

lim_{n \to \infty} \{\frac{S_{T^{(n + N)} \ {o^{'}}_{m}} (j^{m})}{| T^{(n + N)} |} - \frac{1}{| T^{(n - 1)} |} \sum_{k^{m} \in G^{m}} S_{T^{(n - 1)} \ {o^{'}}_{m}} (k^{m}) q^{(N + 1)} (j^{m} | k^{m})\} = 0, P - a . e ., ω \in D (0) .

(71)

where q^(h)(j^m|k^m) is the h th step probability determined by

{\bar{Q}}_{1}

. We have by ergodicity

lim_{N \to \infty} q^{(N + 1)} (j^{m} | k^{m}) = π (j^{m}), \forall k^{m} \in G^{m},

(72)

and $\sum_{k^{m} \in G^{m}} S_{T^{(n - 1)} \ {o^{'}}_{m}} (k^{m}) = | T^{(n - 1)} | - (m - 1)$ . (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} \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q_{k} (0 | X_{1}^{m} (t)), \dots, q_{k} (b - 1 | X_{1}^{m} (t)))] \\ = \sum_{k = 1}^{n} \sum_{t \in L_{k}} H [q (0 | X_{1}^{m} (t)), \dots, q (b - 1 | X_{1}^{m} (t)))] \\ = - \sum_{k = 1}^{n} \sum_{t \in L_{k}} \sum_{j \in G} q (j | X_{1}^{m} (t)) \ln q (j | X_{1}^{m} (t)) \\ = - \sum_{k = 1}^{n} \sum_{t \in L_{k}} \sum_{j \in G} \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 \sum_{k = 0}^{n - 1} \sum_{t \in L_{k}} \sum_{j \in G} \sum_{i^{m} \in G^{m}} I_{i_{1}} (X_{{(m - 1)}_{t}}) \dots I_{i_{m}} (X_{t}) q (j | i^{m}) \ln q (j | i^{m}) \\ = - N \sum_{j \in G} \sum_{i^{m} \in G^{m}} S_{T} (n - 1) \ {o'}_{m} (i^{m}) q (j | i^{m}) \ln q (j | i^{m}) . \end{array}

(73)

Noticing that $lim_{n \to \infty} \frac{| T^{(n - 1)} |}{| T^{(n)} |} = \frac{1}{N}$ , 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^{(n)}} \ o_{m}^{'} (i^{m}), {S_{T}}_{(n)} \ o_{m} (i^{m + 1})

and f_n(ω) be defined as before. Let

Q_{n} = Q_{1} = (q (j | i^{m})), q (j | i^{m}) > 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

lim_{n \to \infty} q_{n} (j | i^{m}) = q (j | i^{m}), \forall i^{m} \in G^{m}, j \in G,

(75)

then

lim_{n \to \infty} \frac{S_{T^{(n)} \ {o^{'}}_{m}} (i^{m})}{| T^{(n)} |} = π (i^{m}), P - a . e . ω \in D (0),

(76)

lim_{n \to \infty} \frac{S_{T^{(n)} \ o_{m}} (i^{m + 1})}{| T^{(n)} |} = π (i^{m}) q (i_{m + 1} | i^{m}), P - a . e . ω \in D (0),

(77)

lim_{n \to \infty} f_{n} (ω) = - \sum_{i^{m} \in G^{m}} \sum_{j \in G} q (j | i^{m}) ln q (j | i^{m}), P - a . e . ω \in D (0),

(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

lim_{n \to \infty} \frac{1}{| T^{(n)} |} \sum_{k = 1}^{n} \sum_{t \in L_{k}} | q_{k} (j | i^{m}) ln q_{k} (j | i^{m}) - q_{k} (j | i^{m}) ln q_{k} (j | i^{m}) |