# 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(σt 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.

Let σ, t(σ, to, -1, - 2,..., - (m - 1)) be vertices of an m rooted Cayler tree T. Write tσ 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, to, -1, - 2,..., - (m - 1)) of tree T, denote by σ t the vertex farthest from o satisfying σ tσ and σ tt.

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(to, -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. XA = {X t , t A} is a stochastic process indexed by a set A, and denoted by |A| the number of vertices of A, xA is the realization of XA.

Let $\left(\mathrm{\Omega },\mathcal{F}\right)$ be a measure space, {X t , tT} 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

$P\left({x}^{{T}^{\left(n\right)}}\right)=P\left({X}^{{T}^{\left(n\right)}}={x}^{{T}^{\left(n\right)}}\right),\phantom{\rule{1em}{0ex}}{x}^{{T}^{\left(n\right)}}\in {G}^{{T}^{\left(n\right)}}.$
(1)

Let

${f}_{n}\left(\omega \right)=-\frac{1}{|{T}^{\left(n\right)}|}\text{ln}P\left({X}^{{T}^{\left(n\right)}}\right)\phantom{\rule{2.77695pt}{0ex}}.$
(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

$Q\left({x}^{{T}^{\left(n\right)}}\right)=Q\left({X}^{{T}^{\left(n\right)}}={x}^{{T}^{\left(n\right)}}\right),\phantom{\rule{1em}{0ex}}{x}^{{T}^{\left(n\right)}}\in {G}^{{T}^{\left(n\right)}}.$
(3)

Let

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

$P=\left(P\left(y|{x}_{1},{x}_{2},\dots ,{x}_{m}\right)\right),\phantom{\rule{1em}{0ex}}P\left(y|{x}_{1},{x}_{2},\dots ,{x}_{m}\right)\ge 0,{x}_{1},{x}_{2},\dots ,{x}_{m},y\in G.$

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

$q=\left(q\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)\right),\phantom{\rule{1em}{0ex}}{x}_{1},{x}_{2},\dots ,{x}_{m}\in G$
(5)

be a distribution on Gm, and

${Q}_{n}=\left({q}_{n}\left(y|{x}_{1},{x}_{2},\dots ,{x}_{m}\right)\right),\phantom{\rule{1em}{0ex}}{x}_{1},{x}_{2},\dots ,{x}_{m},y\in G,n\ge 1$
(6)

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

(7)

and

$\begin{array}{c}Q\left({X}_{-\left(m-1\right)}={x}_{1},\dots ,{X}_{-1}={x}_{m-1},{X}_{o}={x}_{m}\right)\\ =q\left({x}_{1},\dots ,{x}_{m-1},{x}_{m}\right),\phantom{\rule{1em}{0ex}}{x}_{1},\dots ,{x}_{m}\in G,\end{array}$
(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(xT(n)), Q(xT(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 Gm+1. Let ${ℱ}_{n}=\sigma \left({X}^{{T}^{\left(n\right)}}\right)\left(n\ge 1\right)$. 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 Gm+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 im Gm,

${b}_{\alpha }\left({i}^{m}\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{E}_{Q}\left[{e}^{a|{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|}|{X}_{1}^{m}\left(t\right)={i}^{m}\right]\le \tau .$
(13)

Let

${A}_{t}=\frac{2\tau }{{e}^{2}{\left(t-\alpha \right)}^{2}},$
(14)

where o < t < a. Thus, when 0 ≤ ct2A t , we have

$\underset{n\to \infty }{\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{|{T}^{\left(n\right)}|}|\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}|\mathrm{lim}\mathrm{sup}}\le 2\sqrt{c{A}_{t}},\phantom{\rule{0.1em}{0ex}}P-a.e.,\omega \in D\left(c\right).$
(15)

In particular,

$\underset{n\to \infty }{\text{lim}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}=0,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(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

$\underset{n\to \infty }{\text{lim}}\phantom{\rule{2.77695pt}{0ex}}{t}_{n}\left(\lambda ,\omega \right)=t\left(\lambda ,\omega \right)<\infty ,\phantom{\rule{1em}{0ex}}P-a.e.$

Hence,

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\text{ln}{t}_{n}\left(\lambda ,\omega \right)\le 0,\phantom{\rule{1em}{0ex}}P-a.e..$
(17)

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

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\left[\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-\text{ln}{E}_{Q}\left[{e}^{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}|{X}_{1}^{m}\left(t\right)\right]\right\}-\text{ln}\frac{P\left({X}^{{T}^{\left(n\right)}}\right)}{Q\left({X}^{T\left(n\right)}\right)}\right]\le 0,P-a.e.$
(18)

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

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-\text{ln}{E}_{Q}\left[{e}^{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}|{X}_{1}^{m}\left(t\right)\right]\right\}\le c,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).$
(19)

This implies that

$\begin{array}{c}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \le \underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{\text{ln}{E}_{Q}\left[{e}^{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}|{X}_{1}^{m}\left(t\right)\right]-{E}_{Q}\left[\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}+c,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right)\end{array}$
(20)

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

$\text{max}\left\{{x}^{2}{e}^{-hx},x\ge 0\right\}=4{e}^{-2}/{h}^{2}\left(h>0\right).$
(21)

We have

$\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{\mathrm{ln}{E}_{Q}\left[{e}^{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}|{X}_{1}^{m}\left(t\right)\right]-{E}_{Q}\left[\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{0.1em}{0ex}}\le \underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{E}_{Q}\left[{e}^{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}|{X}_{1}^{m}\left(t\right)\right]-1-{E}_{Q}\left[\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{0.1em}{0ex}}\le \frac{{\lambda }^{2}}{2}\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{E}_{Q}\left[{g}_{k}2\left({X}_{0}^{m}\left(t\right)\right){e}^{|\lambda {||}_{{g}_{k}}\left({X}_{0}^{m}\left(t\right)\right)|}|{X}_{1}^{m}\left(t\right)\right]\\ \phantom{\rule{0.1em}{0ex}}=\frac{{\lambda }^{2}}{2}\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{E}_{Q}\left[{e}^{\alpha |{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|}{g}_{k}2\left({X}_{0}^{m}\left(t\right)\right){e}^{\left(|\lambda |-\alpha \right)|{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|}|{X}_{1}^{m}\left(t\right)\right]\\ \phantom{\rule{0.1em}{0ex}}\le \frac{{\lambda }^{2}}{2}\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{E}_{Q}\left[{e}^{\alpha |{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|}4{e}^{-2}/{\left(|\lambda |-a\right)}^{2}|{X}_{1}^{m}\left(t\right)\right]\\ \phantom{\rule{0.1em}{0ex}}\le 2{\lambda }^{2}\tau /{e}^{2}{\left(t-\alpha \right)}^{2}.\end{array}$
(22)

By (20) and (22), we have

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{2.77695pt}{0ex}}\le {\lambda }^{2}{A}_{t}+c,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(23)

When 0 < λ < t < α, we have by (23)

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{2.77695pt}{0ex}}\le \lambda {A}_{t}+c/\lambda ,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(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

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\le 2\sqrt{c{A}_{t}},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).$
(25)

When c = 0, we have by (24)

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\le \lambda {A}_{t},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(26)

Letting λ → 0+ in (26), we obtain

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\le 0,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(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

$\underset{n\to \infty }{\text{lim}\text{inf}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\ge -2\sqrt{c{A}_{t}},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).$
(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},\phantom{\rule{1em}{0ex}}0
(29)

Let f n (ω) be defined by (2). Under the conditions of Theorem 1, when 0 ≤ ct2H t , we have

$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{\text{lim}\text{sup}}\left\{{f}_{n}\left(\omega \right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}\\ \le 2\sqrt{c{H}_{t}},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right),\end{array}$
(30)
$\begin{array}{c}\phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{\text{lim}\text{inf}}\left\{{f}_{n}\left(\omega \right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}\\ \ge -2\sqrt{c{H}_{t}}-c,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right),\end{array}$
(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

$\begin{array}{c}{E}_{Q}\left[{e}^{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}|{X}_{1}^{m}\left(t\right)={i}^{m}\right]\\ \phantom{\rule{1em}{0ex}}=\sum _{j\in G}{e}^{|-\text{ln}{q}_{k}\left(j|{i}^{m}\right)|}{q}_{k}\left(j|{i}^{m}\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{j\in G}{q}_{k}\left(j|{i}^{m}\right)/{q}_{k}\left(j|{i}^{m}\right)\\ \phantom{\rule{1em}{0ex}}=b.\end{array}$
(32)

Hence, im Gm,

${b}_{1}\left({i}^{m}\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{E}_{Q}\left[{e}^{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}|{X}_{1}^{m}\left(t\right)={i}^{m}\right]\le b.$
(33)

Noticing that

$\begin{array}{c}{E}_{Q}\left[-\text{ln}{q}_{k}\left({X}_{t}|{X}_{1}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\\ \phantom{\rule{1em}{0ex}}=-\sum _{j\in G}{q}_{k}\left(j|{X}_{1}^{m}\left(t\right)\right)\text{ln}{q}_{k}\left(j|{X}_{1}^{m}\left(t\right)\right)\\ \phantom{\rule{1em}{0ex}}=H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}.\end{array}$
(34)

When 0 ≤ ct2H t , we have by (34),(29) and (15)

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\left\{\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left(-\text{ln}{q}_{k}\left({X}_{t}|{X}_{1}^{m}\left(t\right)\right)\right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le 2\sqrt{c{H}_{t}},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(35)
$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}\left\{\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left(-\text{ln}{q}_{k}\left({X}_{t}|{X}_{1}^{m}\left(t\right)\right)\right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\ge -2\sqrt{c{H}_{t}},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(36)

By (35), (9) and h(P|Q) ≥ 0,

$\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left\{{f}_{n}\left(\omega \right)-\frac{1}{|{T}^{\left(n\right)}|}\underset{k=1}{\overset{n}{|}}\underset{t\in {L}_{k}}{|}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\\ \phantom{\rule{0.1em}{0ex}}\le \underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left\{-\frac{1}{|{T}^{\left(n\right)}|}\mathrm{ln}P\left({X}^{{T}^{\left(n\right)}}\right)-\frac{1}{|{T}^{\left(n\right)}|}\underset{k=1}{\overset{n}{|}}\underset{t\in {L}_{k}}{|}\left(-\mathrm{ln}{q}_{k}\left({X}_{t}|{X}_{1}^{m}\left(t\right)\right)\right\}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left\{\frac{1}{|{T}^{\left(n\right)}|}\underset{k=1}{\overset{n}{|}}\underset{t\in {L}_{k}}{|}\left(-\mathrm{ln}{q}_{k}\left({X}_{t}|{X}_{1}^{m}\left(t\right)\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}-\frac{1}{|{T}^{\left(n\right)}|}\underset{k=1}{\overset{n}{|}}\underset{t\in {L}_{k}}{|}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\phantom{\rule{0.1em}{0ex}}\dots ,\phantom{\rule{0.1em}{0ex}}{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}\\ \phantom{\rule{0.1em}{0ex}}\le 2\sqrt{c{H}_{t}},\phantom{\rule{0.1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(37)

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

$\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left\{{f}_{n}\left(\omega \right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\\ \phantom{\rule{0.1em}{0ex}}\le \underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left\{-\frac{1}{|{T}^{\left(n\right)}|}\mathrm{ln}P\left({X}^{{T}^{\left(n\right)}}\right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left(-\mathrm{ln}{q}_{k}\left({X}_{t}|{X}_{1}^{m}\left(t\right)\right)\right\}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left\{\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left(-\mathrm{ln}{q}_{k}\left({X}_{t}|{X}_{1}^{m}\left(t\right)\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\phantom{\rule{0.1em}{0ex}}\dots ,\phantom{\rule{0.1em}{0ex}}{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}\\ \phantom{\rule{0.1em}{0ex}}\le 2\sqrt{c{H}_{t}},\phantom{\rule{0.1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(38)

This completes the proof of this theorem. □

Corollary 1 Under the conditions of Theorem 2, we have

$\underset{n\to \infty }{\text{lim}}\left\{{f}_{n}\left(\omega \right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}=0,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(39)

If P << Q, then

$\underset{n\to \infty }{\text{lim}}\left\{{f}_{n}\left(\omega \right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}=0,\phantom{\rule{1em}{0ex}}P-a.e.$
(40)

In particular, if P = Q,

$\underset{n\to \infty }{\text{lim}}\left\{{f}_{n}\left(\omega \right)-\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right]\right\}=0,\phantom{\rule{1em}{0ex}}Q-a.e.$
(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

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}|\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}|\phantom{\rule{2.77695pt}{0ex}}\le M\left(c+2\sqrt{c}\right),\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).$
(42)

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

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{1em}{0ex}}\le \underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{E}_{Q}\left[{e}^{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}-1-\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+c\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(43)

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

${e}^{\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)}-1-\lambda {g}_{k}\left({X}_{0}^{m}\left(t\right)\right)\le \phantom{\rule{2.77695pt}{0ex}}|\lambda |M\left({e}^{|\lambda |M}-1\right).$
(44)

By (43) and (44)

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{\lambda }{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{1em}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}|\lambda |M\left({e}^{|\lambda ||M}-1\right)+c,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(45)

When λ > 0, we have by (45)

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{1em}{0ex}}\le M\left({e}^{\lambda M}-1\right)+c/\lambda ,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(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

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \le M\sqrt{c}+\frac{cM}{\text{log}\left(1+\sqrt{c}\right)}\\ \le M\left(2\sqrt{c}+c\right),\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(48)

When λ < 0, it follows from (45) that

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{1em}{0ex}}\ge -M\left({e}^{\lambda M}-1\right)+c/\lambda \phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(49)

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

$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[{g}_{k}\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\\ \phantom{\rule{1em}{0ex}}\ge -M\sqrt{c}-\frac{cM}{\text{log}\left(1+\sqrt{c}\right)}\\ \phantom{\rule{1em}{0ex}}\ge -M\left(2\sqrt{c}+c\right),\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(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 Gm+1. Let M = max g(y1,..., y m +1). Then when c ≥ 0,

$\underset{n\to \infty }{\text{lim}\text{sup}}\frac{1}{|{T}^{\left(n\right)}|}\left|\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{g\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[g\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}\right|\le M\left(c+2\sqrt{c}\right),\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).$
(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 ${S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}_{1},\dots ,{i}_{m}\right)$ be the number of (i1,..., i m ) in the collection of $\left\{{X}_{0}^{m-1}\left(t\right),t\in {T}^{\left(n\right)}\{o}_{m}^{\prime }\right\}$, that is

${S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}_{1},\dots ,{i}_{m}\right)=\sum _{k=0}^{n}\sum _{t\in {L}_{k}}{I}_{{i}_{1}}\left({X}_{{\left(m-1\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m}}\left({X}_{t}\right),$
(52)

${S}_{{T}^{\left(n\right)}\{o}_{m}}\left({i}_{1},\dots ,{i}_{m},{i}_{m+1}\right)$ be the number of (i1,..., i m , i m +1) in the collection of $\left\{{X}_{0}^{m}\left(t\right),t\in {T}^{\left(n\right)}\{o}_{m}\right\}$, that is

${S}_{{T}^{\left(n\right)}\{o}_{m}}\left({i}_{1},\dots ,{i}_{m},{i}_{m+1}\right)=\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{I}_{{i}_{1}}\left({X}_{{m}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m+1}}\left({X}_{t}\right).$
(53)

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

$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{\text{lim}\text{sup}}|\frac{{S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}_{1},\dots ,{i}_{m}\right)}{|{T}^{\left(n\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in {L}_{k}}{I}_{l}\left({X}_{{\left(m-1\right)}_{t}}\right)\\ .{I}_{{i}_{1}}\left({X}_{{\left(m-2\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{t}\right){q}_{k+1}\left({i}_{m}|l,{i}_{1},\dots ,{i}_{m-1}\right)|\phantom{\rule{2.77695pt}{0ex}}\le c+2\sqrt{c},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(54)
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{\text{lim}\text{sup}}|\frac{{S}_{{T}^{\left(n\right)}\{o}_{m}}\left({i}_{1},\dots ,{i}_{m+1}\right)}{|{T}^{\left(n\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{k=0}^{n-1}\sum _{t\in {L}_{k}}{I}_{{i}_{1}}\left({X}_{{\left(m-1\right)}_{t}}\right)\\ .{I}_{{i}_{2}}\left({X}_{{\left(m-2\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m}}\left({X}_{t}\right){q}_{k+1}\left({i}_{m+1}|{i}_{1},\dots ,{i}_{m}\right)|\phantom{\rule{2.77695pt}{0ex}}\le c+2\sqrt{c},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(c\right).\end{array}$
(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.

$\begin{array}{ll}\hfill \sum _{k=1}^{n}\sum _{t\in {L}_{k}}g\left({X}_{0}^{m}\left(t\right)\right)& =\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{I}_{{i}_{1}}\left({X}_{{\left(m-1\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m}}\left({X}_{t}\right)\phantom{\rule{2em}{0ex}}\\ ={S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}_{1},\dots ,{i}_{m}\right)-{I}_{{i}_{1}}\left({X}_{-\left(m-1\right)}\right)\cdot \cdot \cdot {I}_{{i}_{m}}\left({X}_{o}\right),\phantom{\rule{2em}{0ex}}\end{array}$
(56)

and

$\begin{array}{c}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{E}_{Q}\left[g\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\\ \phantom{\rule{1em}{0ex}}=\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\sum _{{x}_{t}\in G}g\left({X}_{1}^{m}\left(t\right),{x}_{t}\right){q}_{k}\left({x}_{t}|{X}_{1}^{m}\left(t\right)\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\sum _{{x}_{t}\in G}{I}_{{i}_{1}}\left({X}_{{\left(m-1\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{{1}_{t}}\right){I}_{{i}_{m}}\left({x}_{t}\right){q}_{k}\left({x}_{t}|{X}_{1}^{m}\left(t\right)\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{I}_{{i}_{1}}\left({X}_{{\left(m-1\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{{1}_{t}}\right){q}_{k}\left({i}_{m}|{X}_{1}^{m}\left(t\right)\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{l\in G}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}{I}_{l}\left({X}_{{m}_{t}}\right){I}_{{i}_{1}}\left({X}_{{\left(m-1\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{{1}_{t}}\right){q}_{k}\left({i}_{m}|l,{i}_{1},\dots ,{i}_{m-1}\right)\\ \phantom{\rule{1em}{0ex}}=N\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in {L}_{k}}{I}_{l}\left({X}_{{\left(m-1\right)}_{t}}\right){I}_{{i}_{1}}\left({X}_{{\left(m-2\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{t}\right){q}_{k+1}\left({i}_{m}|l,{i}_{1},\dots ,{i}_{m-1}\right).\end{array}$
(57)

Noticing that M = max g(y1,..., y m +1) = 1, $\underset{n\to \infty }{\text{lim}}\frac{|{T}^{\left(n-1\right)}|}{|{T}^{\left(n\right)}|}=\frac{1}{N}$, 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.

$\underset{n\to \infty }{\text{lim}}\frac{1}{|{T}^{\left(n\right)}|}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\left\{g\left({X}_{0}^{m}\left(t\right)\right)-{E}_{Q}\left[g\left({X}_{0}^{m}\left(t\right)\right)|{X}_{1}^{m}\left(t\right)\right]\right\}=0,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right),$
(58)
$\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}\left\{\frac{{S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}_{1},\dots ,{i}_{m}\right)}{|{T}^{\left(n\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in {L}_{k}}{I}_{l}\left({X}_{{\left(m-1\right)}_{t}}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}.{I}_{{i}_{1}}\left({X}_{{\left(m-2\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{t}\right){q}_{k+1}\left({i}_{m}|l,{i}_{1},...,{i}_{m-1}\right)\right\}=0,\phantom{\rule{0.1em}{0ex}}P-a.e.,\omega \in D\left(0\right),\end{array}$
(59)
$\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}\left\{\frac{{S}_{{T}^{\left(n\right)}\{o}_{m}}\left({i}_{1},\dots ,{i}_{m+1}\right)}{|{T}^{\left(n\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{k=0}^{n-1}\sum _{t\in {L}_{k}}{I}_{{i}_{1}}\left({X}_{{\left(m-1\right)}_{t}}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}.{I}_{{i}_{2}}\left({X}_{{\left(m-2\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m}}\left({X}_{t}\right){q}_{k+1}\left({i}_{m+1}|{i}_{1},\dots ,{i}_{m}\right)\right\}=0,\phantom{\rule{0.1em}{0ex}}P-a.e.,\omega \in D\left(0\right).\end{array}$
(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

${Q}_{1}=\left(q\left(j|{i}^{m}\right)\right),\phantom{\rule{1em}{0ex}}j\in G,{i}^{m}\in {G}^{m}$
(61)

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

${\stackrel{̄}{Q}}_{1}=\left(q\left({j}^{m}|{i}^{m}\right)\right),\phantom{\rule{1em}{0ex}}{i}^{m},{j}^{m}\in {G}^{m},$
(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}}\phantom{\rule{2.77695pt}{0ex}}{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 ${\stackrel{̄}{Q}}_{1}$ is called an m-dimensional stochastic matrix determined by the m th-order transition matrix.Q1.

Lemma 2 (see [16]). Let ${\stackrel{̄}{Q}}_{\mathsf{\text{1}}}$ 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

${Q}_{1}=\left(q\left(j|{i}^{m}\right)\right),\phantom{\rule{1em}{0ex}}q\left(j|{i}^{m}\right)>0,\forall j\in G,{i}^{m}\in {G}^{m},$
(64)

then ${\stackrel{̄}{Q}}_{1}$ is ergodic.

Theorem 4 Let {X t , t T} be defined as Theorem 1. Let ${S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}_{1},\dots ,{i}_{m}\right)={{S}_{T}}_{\left(n\right)\{{o}^{\prime }}_{m}}\left({i}^{m}\right),{{S}_{T}}_{\left(n\right)\{o}_{m}}\left({i}_{1},\dots ,{i}_{m},{i}_{m+1}\right)={{S}_{T}}_{\left(n\right)\{o}_{m}}\left({i}^{m+1}\right)$ 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 ${\stackrel{̄}{Q}}_{1}$ determined by Q1 be ergodic. Then for all i1,..., i m +1 G, we have

${\text{lim}}_{n\to \infty }\frac{{S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}^{m}\right)}{|{T}^{\left(n\right)}|}=\pi \left({i}^{m}\right),\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(66)
$\underset{n\to \infty }{\text{lim}}\frac{{{S}_{T}}_{\left(n\right)\{o}_{m}}\left({i}^{m+1}\right)}{|{T}^{\left(n\right)}|}=\pi \left({i}^{m}\right)q\left({i}_{m+1}|{i}^{m}\right),\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(67)
$\underset{n\to \infty }{\text{lim}}{f}_{n}\left(\omega \right)=-\sum _{{i}^{m}\in {G}^{m}}\sum _{j\in G}\pi \left({i}^{m}\right)q\left(j|{i}^{m}\right)\text{ln}q\left(j|{i}^{m}\right),\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(68)

where {π(im), im Gm} is the stationary distribution determined by ${\stackrel{̄}{Q}}_{1}$.

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

$\begin{array}{c}\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in {L}_{k}}{I}_{l}\left({X}_{{\left(m-1\right)}_{t}}\right){I}_{{i}_{1}}\left({X}_{{\left(m-2\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{t}\right){q}_{k+1}\left({i}_{m}|l,{i}_{1},\dots ,{i}_{m-1}\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{l\in G}\sum _{k=0}^{n-1}\sum _{t\in {L}_{k}}{I}_{l}\left({X}_{{\left(m-1\right)}_{t}}\right){I}_{{i}_{1}}\left({X}_{{\left(m-2\right)}_{t}}\right)\cdot \cdot \cdot {I}_{{i}_{m-1}}\left({X}_{t}\right)q\left({i}_{m}|l,{i}_{1},\dots ,{i}_{m-1}\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{l\in G}{S}_{{T}^{\left(n-1\right)}\{{o}^{\prime }}_{m}}\left(l,{i}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{i}_{m-1}\right)q\left({i}_{m}|l,{i}_{1},\dots ,{i}_{m-1}\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{{k}^{m}\in {G}^{m}}{S}_{{T}^{\left(n-1\right)}\{{o}^{\prime }}_{m}}\left({k}^{m}\right)q\left({i}^{m}|{k}^{m}\right).\end{array}$
(69)

By (59) and (69), we have

$\underset{n\to \infty }{\text{lim}}\left\{\frac{{S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}^{m}\right)}{|{T}^{\left(n\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{{k}^{m}\in {G}^{m}}{S}_{T\left(n-1\right)\{{o}^{\prime }}_{m}}\left({k}^{m}\right)q\left({i}^{m}|{k}^{m}\right)\right\}=0,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(70)

Multiplying (70) by q(jm|im), adding them together for im Gm, and using (70) once again, we have

$\begin{array}{ll}\hfill 0& =\sum _{{i}^{m}\in {G}^{m}}q\left({j}^{m}|{i}^{m}\right)\cdot \underset{n\to \infty }{\text{lim}}\left\{\frac{{S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}^{m}\right)}{|{T}^{\left(n\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{{k}^{m}\in {G}^{m}}{S}_{{T}^{\left(n-1\right)}\{{o}^{\prime }}_{m}}\left({k}^{m}\right)q\left({i}^{m}|{k}^{m}\right)\right\}\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim}}\left\{\sum _{{i}^{m}\in {G}^{m}}\frac{{S}_{{T}^{\left(n\right)}\{{o}^{\prime }}_{m}}\left({i}^{m}\right)}{|{T}^{\left(n\right)}|}q\left({j}^{m}|{i}^{m}\right)-\frac{{S}_{{T}^{\left(n+1\right)}\{{o}^{\prime }}_{m}}\left({j}^{m}\right)}{|{T}^{\left(n+1\right)}|}\right\}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}+\underset{n\to \infty }{\text{lim}}\left\{\frac{{S}_{{T}^{\left(n+1\right)}\{{o}^{\prime }}_{m}}\left({j}^{m}\right)}{|{T}^{\left(n+1\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{{k}^{m}\in {G}^{m}}{S}_{{T}^{\left(n-1\right)}\{{o}^{\prime }}_{m}}\left({k}^{m}\right)\sum _{{i}^{m}\in {G}^{m}}q\left({j}^{m}|{i}^{m}\right)q\left({i}^{m}|{k}^{m}\right)\right\}\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim}}\left\{\frac{{S}_{{T}^{\left(n+1\right)}\{{o}^{\prime }}_{m}}\left({j}^{m}\right)}{|{T}^{\left(n+1\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{{k}^{m}\in {G}^{m}}{S}_{{T}^{\left(n-1\right)}\{{o}^{\prime }}_{m}}\left({k}^{m}\right){q}^{\left(2\right)}\left({j}^{m}|{k}^{m}\right)\right\},\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).\phantom{\rule{2em}{0ex}}\end{array}$

By induction, we have

$\underset{n\to \infty }{\text{lim}}\left\{\frac{{S}_{{T}^{\left(n+N\right)}\{{o}^{\prime }}_{m}}\left({j}^{m}\right)}{|{T}^{\left(n+N\right)}|}-\frac{1}{|{T}^{\left(n-1\right)}|}\sum _{{k}^{m}\in {G}^{m}}{S}_{{T}^{\left(n-1\right)}\{{o}^{\prime }}_{m}}\left({k}^{m}\right){q}^{\left(N+1\right)}\left({j}^{m}|{k}^{m}\right)\right\}=0,\phantom{\rule{1em}{0ex}}P-a.e.,\omega \in D\left(0\right).$
(71)

where q(h)(jm|km) is the h th step probability determined by ${\stackrel{̄}{Q}}_{1}$. We have by ergodicity

$\underset{N\to \infty }{\text{lim}}{q}^{\left(N+1\right)}\left({j}^{m}|{k}^{m}\right)=\pi \left({j}^{m}\right),\phantom{\rule{1em}{0ex}}\forall {k}^{m}\in {G}^{m},$
(72)

and ${\sum }_{{k}^{m}\in {G}^{m}}{S}_{{T}^{\left(n-1\right)}\{{o}^{\prime }}_{m}}\left({k}^{m}\right)=\phantom{\rule{2.77695pt}{0ex}}|{T}^{\left(n-1\right)}|-\left(m-1\right)$. (66) follows from (71) and (72). By (66) and (60), Equation 67 follows easily.

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

$\begin{array}{l}\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[{q}_{k}\left(0|{X}_{1}^{m}\left(t\right)\right),\dots ,{q}_{k}\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right)\right]\\ \phantom{\rule{0.1em}{0ex}}=\sum _{k=1}^{n}\sum _{t\in {L}_{k}}H\left[q\left(\text{0}|{X}_{1}^{m}\left(t\right)\right),\dots ,q\left(b-1|{X}_{1}^{m}\left(t\right)\right)\right)\right]\\ \phantom{\rule{0.1em}{0ex}}=-\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\sum _{j\in G}q\left(j|{X}_{1}^{m}\left(t\right)\right)\mathrm{ln}q\left(j|{X}_{1}^{m}\left(t\right)\right)\\ \phantom{\rule{0.1em}{0ex}}=-\sum _{k=1}^{n}\sum _{t\in {L}_{k}}\sum _{j\in G}\sum _{{i}^{m}\in {G}^{m}}{I}_{}\end{array}$