﻿ 多重随机环境中马氏链及其强大数定律
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 浙江大学学报(理学版)  2017, Vol. 44 Issue (4): 411-416  DOI:10.3785/j.issn.1008-9497.2017.04.005 0

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FEI Shilong. The multiple Markov chains in a random environment and the strong law of large numbers[J]. Journal of Zhejiang University(Science Edition), 2017, 44(4): 411-416. DOI: 10.3785/j.issn.1008-9497.2017.04.005.
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### 文章历史

The multiple Markov chains in a random environment and the strong law of large numbers
FEI Shilong
School of Mathematics and Statistics, Suzhou University, Suzhou 234000, Anhui Province, China
Abstract: The model of multiple Markov chains in a random environment is introduced which is a promotion of Markov chains in a random environment with a more general application scope. Two application backgrounds of the multiple Markov chains in a random environment are given. Then, we discuss some relations and properties of the order m Markov chains and the order k Markov chains in a random environment, Markov chains, and 2m dimensional chains. At last, using the property of the multiple Markov chains in a random environment, we obtain the sufficient condition of the strong law of large numbers of the multiple Markov chains in a random environment, which are a promotion of the results from some literatures.
Key words: random environments    Markov chains with order m    strong law of large numbers
0 引言

1 m重MCRE模型及性质

N=Z+为非负整数集, (Ω, $\mathscr{F}$, P)是一概率空间, (Θ, $\mathscr{B}$)与($\mathscr{X}$, $\mathscr{A}$)为任意2个可测空间, $\vec \xi$={ξn, n=0, 1, …}和$\vec X$={Xn, n=0, 1, 2, …}是概率空间(Ω, $\mathscr{F}$, P)上分别取值于Θ$\mathscr{X}$的2个随机序列, (Θm, $\mathscr{B}$m)、($\mathscr{X}$m, $\mathscr{A}$m)分别为(Θ, $\mathscr{B}$)与($\mathscr{X}$, $\mathscr{A}$)的m个乘积可测空间, $\vec \xi$nn+m-1=(ξn, ξn+1, …, ξn+m-1)和$\vec X$nn+m-1=(Xn, Xn+1, …, Xn+m-1)分别为(Θm, $\mathscr{B}$m)与($\mathscr{X}$m, $\mathscr{A}$m)上的2个m维可测随机向量序列, Θm, $\mathscr{X}$m中的元素分别用θm, xm表示.

(ⅰ)对任意θmΘmxm$\mathscr{X}$m, p(θm; xm, ·)是$\mathscr{A}$上的概率测度.

(ⅱ)对任意θmΘmA$\mathscr{A}$, p(θm; ·, A)关于$\mathscr{A}$m可测.

(ⅲ)对任意xm$\mathscr{X}$mA$\mathscr{A}$, p(·;xm, A)关于$\mathscr{B}$m可测.

(ⅳ)对任意A$\mathscr{A}$p(·;·, A)关于$\mathscr{B}$m×$\mathscr{A}$m可测.则称p(·;·, ·)为一个m重随机马尔科夫核.

 $P\left( {{X_{n + m}} \in A\left| {\vec X_0^{n + m - 1},\vec \xi } \right.} \right) = p\left( {\vec \xi _n^{n + m - 1};\vec X_n^{n + m - 1},A} \right),$ (1)
 $P\left( {\vec X_0^{m - 1} \in \vec A_0^{m - 1}\left| {\vec \xi } \right.} \right) = p\left( {\vec X_0^{m - 1} \in \vec A_0^{m - 1}\left| {\vec \xi _0^{m - 1}} \right.} \right),$ (2)

 $\begin{array}{l} p\left( {\left( {{\theta _n}, \cdots ,{\theta _{n + k - 1}}} \right);\left( {{x_n}, \cdots ,{x_{n + k - 1}}} \right);A} \right) = \\ \;\;\;\;\;\;\;\;p\left( {\left( {{\theta _{n + k - m}}, \cdots ,{\theta _{n + k - 1}}} \right);\left( {{x_{n + k - m}}, \cdots ,{x_{n + k - 1}}} \right);A} \right), \end{array}$

 $\begin{array}{l} P\left( {{X_{n + m + 1}} \in A\left| {\vec X_0^{n + m},\vec \xi } \right.} \right) = p\left( {\vec \xi _{n + 1}^{n + m};\vec X_{n + 1}^{n + m},A} \right) = \\ \;\;\;\;\;\;\;P\left( {{X_{n + m + 1}} \in A\left| {\vec X_{n + 1}^{n + m},\vec \xi _{n + 1}^{n + m}} \right.} \right) = \\ \;\;\;\;\;\;\;E\left( {{I_{\left\{ {{X_{n + m + 1}} \in A} \right\}}}\left| {\vec X_{n + 1}^{n + m},\vec \xi _{n + 1}^{n + m}} \right.} \right) = \\ \;\;\;\;\;\;\;E\left( {E\left( {{I_{\left\{ {{X_{n + m + 1}} \in A} \right\}}}\left| {\vec X_{n + 1}^{n + m},\vec \xi _{n + 1}^{n + m}} \right.} \right)\left| {\vec X_n^{n + m},\vec \xi _n^{n + m}} \right.} \right) = \\ \;\;\;\;\;\;\;E\left( {E\left( {{I_{\left\{ {{X_{n + m + 1}} \in A} \right\}}}\left| {\vec X_0^{n + m},\vec \xi } \right.} \right)\left| {\vec X_n^{n + m},\vec \xi _n^{n + m}} \right.} \right) = \\ \;\;\;\;\;\;\;E\left( {{I_{\left\{ {{X_{n + m + 1}} \in A} \right\}}}\left| {\vec X_n^{n + m},\vec \xi _n^{n + m}} \right.} \right) = \\ \;\;\;\;\;\;\;p\left( {\vec X_n^{n + m};\vec \xi _n^{n + m},A} \right). \end{array}$
 $\begin{array}{l} P\left( {{X_m} \in {A_m},\vec X_0^{m - 1} \in \vec A_0^{m - 1}\left| {\vec \xi } \right.} \right) = \\ \;\;\;\;\;\;\;\;E\left( {E\left( {{I_{\left\{ {{X_m} \in {A_m},\vec X_0^{m - 1} \in \vec A_0^{m - 1}} \right\}}}\left| {\vec \xi ,\vec X_0^{m - 1}} \right.} \right)\left| {\vec \xi } \right.} \right) = \\ \;\;\;\;\;\;\;\;E\left( {{I_{\left\{ {\vec X_0^{m - 1} \in \vec A_0^{m - 1}} \right\}}}E\left( {{I_{\left\{ {{X_m} \in {A_m}} \right\}}}\left| {\vec \xi ,\vec X_0^{m - 1}} \right.} \right)\left| {\vec \xi } \right.} \right) = \\ \;\;\;\;\;\;\;\;E\left( {{I_{\left\{ {\vec X_0^{m - 1} \in \vec A_0^{m - 1}} \right\}}}p\left( {\vec \xi _0^{m - 1};\vec X_0^{m - 1},A} \right)\left| {\vec \xi } \right.} \right) = \\ \;\;\;\;\;\;\;\;E\left( {{I_{\left\{ {\vec X_0^{m - 1} \in \vec A_0^{m - 1}} \right\}}}p\left( {\vec \xi _0^{m - 1};\vec X_0^{m - 1},A} \right)\left| {\vec \xi _0^{m - 1}} \right.} \right) = \\ \;\;\;\;\;\;\;\;P\left( {{X_m} \in {A_m},\vec X_0^{m - 1} \in \vec A_0^{m - 1}\left| {\vec \xi _0^{m - 1}} \right.} \right) = \\ \;\;\;\;\;\;\;\;E\left( {E\left( {{I_{\left\{ {{X_m} \in {A_m},\vec X_0^{m - 1} \in \vec A_0^{m - 1}} \right\}}}\left| {\vec \xi _0^{m - 1}} \right.} \right)\left| {\vec \xi _0^m} \right.} \right) = \\ \;\;\;\;\;\;\;\;E\left( {E\left( {{I_{\left\{ {{X_m} \in {A_m},\vec X_0^{m - 1} \in \vec A_0^{m - 1}} \right\}}}\left| {\vec \xi } \right.} \right)\left| {\vec \xi _0^m} \right.} \right) = \\ \;\;\;\;\;\;\;\;E\left( {{I_{\left\{ {{X_m} \in {A_m},\vec X_0^{m - 1} \in \vec A_0^{m - 1}} \right\}}}\left| {\vec \xi _0^m} \right.} \right) = \\ \;\;\;\;\;\;\;\;P\left( {{X_m} \in {A_m},\vec X_0^{m - 1} \in \vec A_0^{m - 1}\left| {\vec \xi _0^m} \right.} \right). \end{array}$

 $P\left( {\vec X_0^m \in \vec A_0^m\left| {\vec \xi } \right.} \right) = p\left( {\vec X_0^m \in \vec A_0^m\left| {\vec \xi _0^m} \right.} \right).$

$\vec X$m+1重随机环境$\vec \xi$中的马尔科夫链.由数学归纳法易证$\vec X$k重随机环境$\vec \xi$中的马尔科夫链.

(a) ($\vec X$, $\vec \xi$)为(Ω, $\mathscr{F}$, P)上的m重MCRE;

(b)下列条件成立:

 $\begin{array}{l} P\left( {{X_{n + m}} \in A,{\xi _{n + m}} \in B\left| {\vec X_0^{n + m - 1},\vec \xi _0^{n + m - 1}} \right.} \right) = \\ \;\;\;\;\;\;\;\;p\left( {\vec \xi _n^{n + m - 1};\vec X_n^{n + m - 1},A} \right) \cdot P\left( {{\xi _{n + m}} \in B\left| {\vec \xi _0^{n + m - 1}} \right.} \right); \end{array}$

(c)下列条件成立:

 $\begin{array}{l} P\left( {{X_{n + m}} \in A,\vec \xi _{n + m}^\infty \in \vec B_{n + m}^\infty \left| {\vec X_0^{n + m - 1},\vec \xi _0^{n + m - 1}} \right.} \right) = \\ \;\;\;\;\;\;\;\;p\left( {\vec \xi _n^{n + m - 1};\vec X_n^{n + m - 1},A} \right) \cdot P\left( {\vec \xi _{n + m}^\infty \in \vec B_{n + m}^\infty \left| {\vec \xi _0^{n + m - 1}} \right.} \right). \end{array}$

(1) 双链$\vec Y$={Yn=(Xn, ξn), n≥0}为l重马尔科夫链, 其中, l=max{m, k}.

(2) {Wn=(Xn, Xn+1, …, Xn+l-1, ξn, ξn+1, …, ξn+l-1), n≥0}为马尔科夫链.

 $\begin{array}{l} P\left( {\left( {{X_{n + l}},{\xi _{n + l}}} \right) \in A \times B\left| {\left( {{X_0},{\xi _0}} \right), \cdots ,\left( {{X_{n + l - 1}},{\xi _{n + l - 1}}} \right)} \right.} \right) = \\ \;\;\;\;\;\;\;\;\;P\left( {\left( {{X_{n + l}} \in A,{\xi _{n + l}} \in B} \right)\left| {\vec X_0^{n + l - 1},\vec \xi _0^{n + l - 1}} \right.} \right) = \\ \;\;\;\;\;\;\;\;\;E\left( {P\left( {{X_{n + l}} \in A,{\xi _{n + l}} \in B\left| {\vec X_0^{n + l - 1},\vec \xi _0^{n + l - 1}} \right.} \right) \times } \right.\\ \;\;\;\;\;\;\;\;\;\left. {\left| {\vec X_0^{n + l - 1},\vec \xi _0^{n + l - 1}} \right.} \right) = E\left( {p\left( {\vec \xi _0^{n + l - 1};\vec X_0^{n + l - 1},A} \right) \times } \right.\\ \;\;\;\;\;\;\;\;\;\left. {P\left( {{\xi _{n + 1}} \in B\left| {\vec \xi _0^{n + l - 1}} \right.} \right)\left| {\vec X_0^{n + l - 1},\vec \xi _0^{n + l - 1}} \right.} \right) = \\ \;\;\;\;\;\;\;\;\;E\left( {p\left( {\vec \xi _n^{n + l - 1};\vec X_n^{n + l - 1},A} \right)P\left( {{\xi _{n + 1}} \in B\left| {\vec \xi _n^{n + l - 1}} \right.} \right)\left| {} \right.} \right.\\ \;\;\;\;\;\;\;\;\;\left. {\vec X_0^{n + l - 1},\vec \xi _0^{n + l - 1}} \right) = E\left( {p\left( {\vec \xi _n^{n + l - 1};\vec X_n^{n + l - 1},A} \right) \times } \right.\\ \;\;\;\;\;\;\;\;\;\left. {P\left( {{\xi _{n + 1}} \in B\left| {\vec \xi _n^{n + l - 1}} \right.} \right)\left| {\vec X_n^{n + l - 1},\vec \xi _n^{n + l - 1}} \right.} \right) = \\ \;\;\;\;\;\;\;\;\;P\left( {\left( {{X_{n + l}},{\xi _{n + l}}} \right) \in A \times B} \right)\left| {} \right.\\ \;\;\;\;\;\;\;\;\;\left( {\left( {{X_{n + l - 1}},{\xi _{n + l - 1}}} \right), \cdots ,\left( {{X_n},{\xi _n}} \right)} \right). \end{array}$

(2) 令Wn=(Yn, Yn+1, …, Yn+l-1), 则由(1) 知{Yn=(Xn, ξn), n≥0}为一个l重马尔科夫链, 从而

 $\left\{ {{W_n} = \left( {{Y_n},{Y_{n + 1}}, \cdots ,{Y_{n + l - 1}}} \right),n \ge 0} \right\}$

(1) 双链$\vec Y$={Yn=(Xn, ξn), n≥0}为m重马尔科夫链.

(2) 2m维链{Wn=(Yn, Yn+1, …, Yn+m-1), n≥0}为马尔科夫链.

(1) 双链$\vec Y$={Yn=(Xn, ξn), n≥0}为时齐的m重马尔科夫链.

(2) 2m维链{Wn=(Yn, Yn+1, …, Yn+m-1), n≥0}为时齐的马尔科夫链.

2 m重MCRE的强大数定律

 $\begin{array}{l} P\left( {\left( {{Y_n},{Y_{n + 1}}, \cdots ,{Y_{n + m - 1}}} \right)} \right. = \\ \;\;\;\;\;\;\left. {\left( {{x_0},{\theta _0},{x_1},{\theta _1}, \cdots ,{x_{m - 1}},{\theta _{m - 1}}} \right),{\rm{i}},{\rm{o}}} \right) = 1 \end{array}$ (1)

 $\begin{array}{*{20}{c}} {{\tau _{k + 1}} = \inf \left\{ {n > {\tau _k}:\left( {{Y_n},{Y_{n + 1}}, \cdots ,{Y_{n + m - 1}}} \right) = } \right.}\\ {\left. {\left( {{x_0},{\theta _0},{x_1},{\theta _1}, \cdots ,{x_{m - 1}},{\theta _{m - 1}}} \right)} \right\},} \end{array}$

 $\begin{array}{*{20}{c}} {{\sigma _k} = {\tau _{k + 1}} - {\tau _k},\;\;{\Omega _{{\tau _k}}} = \left\{ {{\tau _k} < \infty } \right\},}\\ {{\mathscr{F}_n} = \sigma \left( {\left( {{X_i},{\xi _i}} \right):0 \le i \le n + m - 1} \right),}\\ {{\mathscr{F}_{{\tau _k}}} = \left\{ {A \subset {\Omega _{{\tau _k}}}:\forall n \ge 0,A \cap \left\{ {{\tau _k} \le n} \right\} \in {\mathscr{F}_{\rm n}}} \right\},}\\ {{\mathscr{F}^{{\tau _k}}} = \sigma \left( {\left( {{X_{{\tau _k} + n}},{\xi _{{\tau _k} + n}}} \right):n \ge 0} \right).} \end{array}$

 $\begin{array}{l} {Z_k} = \sum\limits_{i = {\tau _k}}^{{\tau _{k + 1}} - 1} {{f_i}\left( {{X_i},{\xi _i},{X_{i + 1}},{\xi _{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _{i + m - 1}}} \right)} = \\ \;\;\;\;\;\;\;\;\sum\limits_{i = {\tau _k}}^{{\tau _{k + 1}} - 1} {{f_i}\left( {{Y_i},{Y_{i + 1}}, \cdots ,{Y_{i + m - 1}}} \right)} , \end{array}$
 $\begin{array}{l} {U_k} = \sum\limits_{i = {\tau _k}}^{{\tau _{k + 1}} - 1} {\left| {{f_i}\left( {{X_i},{\xi _i},{X_{i + 1}},{\xi _{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _{i + m - 1}}} \right)} \right|} = \\ \;\;\;\;\;\;\;\;\sum\limits_{i = {\tau _k}}^{{\tau _{k + 1}} - 1} {\left| {{f_i}\left( {{Y_i},{Y_{i + 1}}, \cdots ,{Y_{i + m - 1}}} \right)} \right|} , \end{array}$
 $\begin{array}{l} {{Z'}_k} = \sum\limits_{i = {\tau _{{l_{\left( n \right)}}}}}^k {{f_i}\left( {{X_i},{X_{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _i},{\xi _{i + 1}}, \cdots ,{\xi _{i + m - 1}}} \right)} = \\ \;\;\;\;\;\;\;\;\sum\limits_{i = {\tau _{{l_{\left( n \right)}}}}}^k {{f_i}\left( {{Y_i},{Y_{i + 1}}, \cdots ,{Y_{i + m - 1}}} \right)} , \end{array}$

 $\begin{array}{l} \sum\limits_{i = 1}^n {{f_i}\left( {{X_i},{X_{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _i},{\xi _{i + 1}}, \cdots ,{\xi _{i + m - 1}}} \right)} = \\ \;\;\;\;\;\;\;\;\sum\limits_{i = 1}^{l\left( n \right) - 1} {{Z_i} + {{Y'}_n}} . \end{array}$

(ⅱ)若EZk有限, ∀k≥0, 则

 $\int_{\left\{ {l\left( n \right) < m} \right\}} {{Z_m}{\rm{d}}P} = \left( {E{Z_m}} \right)P\left( {l\left( n \right) < m} \right).$

 $\begin{array}{l} \left\{ {{Z_r} = {c_r},{\tau _k} = n} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\bigcup\limits_{r \le j < l \le n - k + r + 1} {\left\{ {{Z_r} = {c_r},{\tau _r} = j,{\tau _{r + 1}} = l,{\tau _k} = n} \right\} = } \\ \;\;\;\;\;\;\;\;\;\;\;\bigcup\limits_{r \le j < l \le n - k + r + 1} {\left\{ {\sum\limits_{i = j}^{l - 1} {{f_i}\left( {{Y_i},{Y_{i + 1}}, \cdots ,{Y_{i + m - 1}}} \right)} = {c_r},{\tau _r} = } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\left. {j,{\tau _{r + 1}} = l,{\tau _k} = n} \right\} \in {\mathscr{F}_n}. \end{array}$

 $\left\{ {{\sigma _k} = u} \right\} = \left\{ {{W_{{\tau _k} + 1}} \ne w, \cdots ,{W_{{\tau _k} + u - 1}} \ne w,{W_{{\tau _k} + u}} = w} \right\} \in {\mathscr{F}^{{\tau _k}}}.$

 $\begin{array}{l} P\left( {{Z_0} = {c_0},{Z_1} = {c_1},{Z_{k - 1}} = {c_{k - 1}},{Z_k} = {c_k}} \right) = \\ \;\;\;\;\;\;\;\;\sum\limits_{u = 1}^\infty {\sum\limits_{n = k}^\infty {P\left( {{Z_0} = {c_0},{Z_1} = {c_1},{Z_{k - 1}} = {c_{k - 1}},{\tau _k} = n,} \right.} } \\ \;\;\;\;\;\;\;\;\left. {{\sigma _k} = u,\sum\limits_{i = {\tau _k}}^{{\tau _k} + u - 1} {{f_i}\left( {{W_i}} \right) = {c_k}\left| {{\mathscr{F}_{{\tau _k}}}} \right.} } \right) = \\ \;\;\;\;\;\;\;\;\sum\limits_{u = 1}^\infty {\sum\limits_{n = k}^\infty {\int_{\left\{ {{Z_0} = {c_0},{Z_1} = {c_1},{Z_{k - 1}} = {c_{k - 1}},{\tau _k} = n} \right\}} {P\left( {{\sigma _k} = u,} \right.} } } \\ \;\;\;\;\;\;\;\;\left. {\sum\limits_{i = {\tau _k}}^{{\tau _k} + u - 1} {{f_i}\left( {{W_i}} \right) = {c_k}\left| {{W_{{\tau _k}}}} \right.} } \right) = \\ \;\;\;\;\;\;\;\;\sum\limits_{u = 1}^\infty {\sum\limits_{n = k}^\infty {P\left( {{Z_0} = {c_0},{Z_1} = {c_1},{Z_{k - 1}} = {c_{k - 1}},{\tau _k} = n} \right)} } \times \\ \;\;\;\;\;\;\;\;P\left( {{\sigma _k} = u,\sum\limits_{i = {\tau _k}}^{{\tau _k} + u - 1} {{f_i}\left( {{W_i}} \right) = {c_k}} } \right) = \\ \;\;\;\;\;\;\;\;P\left( {{Z_0} = {c_0},{Z_1} = {c_1},{Z_{k - 1}} = {c_{k - 1}}} \right)P\left( {{Z_k} = {c_k}} \right). \end{array}$

(ⅱ){l(n)＜m}={nτm}={τmn}c$\mathscr{F}$τm, 从而

 $\begin{array}{l} \int_{\left\{ {l\left( n \right) < m} \right\}} {{Z_m}{\rm{d}}P} = \int_{\left\{ {l\left( n \right) < m} \right\}} {E\left( {{Z_m}\left| {{\mathscr{F}_{{\tau _m}}}} \right.} \right){\rm{d}}P} = \\ \;\;\;\;\;\;\;\int_{l\left( n \right) < m} {E\left( {{Z_m}\left| {{W_{{\tau _m}}}} \right.} \right){\rm{d}}P} = \\ \;\;\;\;\;\;\;\int_{\left\{ {l\left( n \right) < m} \right\}} {E{Z_m}{\rm{d}}P} = P\left( {E{Z_m}} \right)P\left( {l\left( n \right) < m} \right). \end{array}$

 $\begin{array}{*{20}{c}} {{q_n}\left( x \right) \equiv P\left( {{U_n}\left( x \right) + {\sigma _n}\left( x \right) > x} \right) \le }\\ {q\left( x \right) \equiv 1 - {F_Y}\left( x \right).} \end{array}$

(1) E|Zn|+nEUn+nEY＜∞;

(2) $E\mathop {\max }\limits_{0 \le i \le n} {U_i}$＜∞;

(3) $\frac{1}{{n + 1}}E\mathop {\max }\limits_{0 \le i \le n} {U_i} \to 0$(n→∞);

(1) $\frac{1}{n}\sum\limits_{k = 0}^{n -1} {\left({{Z_k} -E{Z_k}} \right)} \to 0$(n→∞) a.s;

(2) $\frac{1}{n}\sum\limits_{k = 0}^{n -1} {\left({{U_k} -E{U_k}} \right)} \to 0$(n→∞) a.s;

(3) $\frac{1}{n}\sum\limits_{k = 0}^{n -1} {\left({{\sigma _k} -E{\sigma _k}} \right)} \to 0$(n→∞) a.s.

 $\frac{1}{n}E\left[ {\sum\limits_{k = 0}^{l\left( n \right) - 1} {{Z_k}} - \sum\limits_{k = 0}^{l\left( n \right) - 1} {E{Z_k}} } \right] \to 0\left( {n \to \infty } \right)\;{\rm{a}}.\;{\rm{s}}.$

 $\begin{array}{l} E\left[ {\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right)} } \right] = \\ \;\;\;\;\;\;\;\;\sum\limits_{m = 1}^{n - 1} {\sum\limits_{k = 0}^{m - 1} {\int_{\left\{ {l\left( n \right) = m} \right\}} {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } } = \\ \;\;\;\;\;\;\;\;\sum\limits_{k = 0}^{n - 2} {\sum\limits_{m = k + 1}^{n - 1} {\int_{\left\{ {l\left( n \right) = m} \right\}} {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } } = \\ \;\;\;\;\;\;\;\;\sum\limits_{k = 0}^{n - 2} {\int_{\left\{ {l\left( n \right) > k} \right\}} {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } = \\ \;\;\;\;\;\;\;\;\sum\limits_{k = 0}^{n - 1} {\int_\Omega {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } - \sum\limits_{k = 0}^{n - 1} {\int_{\left\{ {l\left( n \right) \le k} \right\}} {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } = \\ \;\;\;\;\;\;\;\;\sum\limits_{k = 0}^{n - 1} {\int_\Omega {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } - \sum\limits_{k = 0}^{n - 1} {\int_{\left\{ {l\left( n \right) < k} \right\}} {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } - \\ \;\;\;\;\;\;\;\;\sum\limits_{k = 0}^{n - 1} {\int_{\left\{ {l\left( n \right) = k} \right\}} {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } . \end{array}$

 $E\left[ {\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right)} } \right]\sum\limits_{k = 0}^{n - 1} {\int_{\left\{ {l\left( n \right) = k} \right\}} {\left( {{Z_k} - E{Z_k}} \right){\rm{d}}P} } .$

 $\begin{array}{l} \left| {E\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right)} } \right| = \left| {\sum\limits_{k = 0}^{n - 1} {\int_{\left\{ {l\left( n \right) = k} \right\}} {\left( {E{Z_k} - {Z_k}} \right){\rm{d}}P} } } \right| \le \\ \;\;\;\;\;\sum\limits_{k = 0}^{n - 1} {\int_{\left\{ {l\left( n \right) = k} \right\}} {\left( {E\left| {{Z_k}} \right| + \left| {{Z_k}} \right|} \right){\rm{d}}P} } \le \\ \;\;\;\;\;\sum\limits_{k = 0}^{n - 1} {\int_{\left\{ {l\left( n \right) = k} \right\}} {\left( {E\mathop {\max }\limits_{0 \le i \le n - 1} {U_i} + \mathop {\max }\limits_{0 \le i \le n - 1} {U_i}} \right){\rm{d}}P} } = \\ \;\;\;\;\;2E\left( {\mathop {\max }\limits_{0 \le i \le n - 1} {U_i}} \right). \end{array}$

 $\frac{1}{n}E\left[ {\sum\limits_{k = 0}^{l\left( n \right) - 1} {{Z_k}} - \sum\limits_{k = 0}^{l\left( n \right) - 1} {E{Z_k}} } \right] \to 0\left( {n \to \infty } \right)\;{\rm{a}}.\;{\rm{s}}.$

 $\frac{1}{n}\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right) \to 0\left( {n \to \infty } \right)\;{\rm{a}}.\;{\rm{s}}.}$

 $\begin{array}{l} \frac{1}{n}\left| {\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right)} } \right| = \frac{{l\left( n \right)}}{n} \cdot \frac{1}{{l\left( n \right)}}\left| {\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right)} } \right| \le \\ \;\;\;\frac{1}{{l\left( n \right)}}\left| {\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right)} } \right| \to 0\;{\rm{a}}.\;{\rm{s}}. \end{array}$

 $\frac{1}{n}\sum\limits_{k = 0}^{l\left( n \right) - 1} {\left( {{Z_k} - E{Z_k}} \right) \to 0\left( {n \to \infty } \right)\;{\rm{a}}.\;{\rm{s}}.}$

 $\begin{array}{l} \frac{1}{n}\left[ {\sum\limits_{i = 0}^{n - 1} {\left( {{f_i}\left( {{X_i},{\xi _i},{X_{i + 1}},{\xi _{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _{i + m - 1}}} \right) - } \right.} } \right.\\ \;\;\;\;\;\;\left. {\left. {E{f_i}\left( {{X_i},{\xi _i},{X_{i + 1}},{\xi _{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _{i + m - 1}}} \right)} \right)} \right] \to \\ \;\;\;\;\;\;0\left( {n \to \infty } \right)\;{\rm{a}}.\;{\rm{s}}. \end{array}$

 $\begin{array}{l} \frac{1}{n}\left[ {\sum\limits_{i = 0}^{n - 1} {\left( {{f_i}\left( {{X_i},{\xi _i},{X_{i + 1}},{\xi _{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _{i + m - 1}}} \right) - } \right.} } \right.\\ \;\;\;\;\;\;\left. {\left. {E{f_i}\left( {{X_i},{\xi _i},{X_{i + 1}},{\xi _{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _{i + m - 1}}} \right)} \right)} \right] = \\ \;\;\;\;\;\;\frac{1}{n}\left[ {\sum\limits_{i = 0}^{n - 1} {{f_i}\left( {{Y_i},{Y_{i + 1}}, \cdots ,{Y_{i + m - 1}}} \right)} - \sum\limits_{i = 0}^{n - 1} {E{f_i}\left( {{Y_i},} \right.} } \right.\\ \;\;\;\;\;\;\left. {\left. {{Y_{i + 1}}, \cdots ,{Y_{i + m - 1}}} \right)} \right] \le \frac{1}{n}\left| {\sum\limits_{i = 0}^{l\left( n \right) - 1} {{Z_i}} - \sum\limits_{i = 0}^{l\left( n \right) - 1} {E{Z_i}} } \right| + \\ \;\;\;\;\;\;\frac{1}{n}\left| {\sum\limits_{i = 0}^{l\left( n \right) - 1} {E{Z_i}} - \sum\limits_{i = 0}^{l\left( n \right) - 1} {{Z_i}} } \right| + \frac{1}{n}\left| {{{Z'}_n}} \right| + \frac{1}{n}\left| {E{{Z'}_n}} \right|, \end{array}$

 $\frac{1}{n}\left[ {\sum\limits_{i = 0}^{n - 1} {{g_i}\left( {{X_i},{\xi _i}} \right)} - E{g_i}\left( {{X_i},{\xi _i}} \right)} \right] \to 0\left( {n \to \infty } \right)\;{\rm{a}}.\;{\rm{s}}.$

 ${f_i}\left( {{X_i},{\xi _i},{X_{i + 1}},{\xi _{i + 1}}, \cdots ,{X_{i + m - 1}},{\xi _{i + m - 1}}} \right) = {g_i}\left( {{X_i},{\xi _i}} \right).$

 [1] COGBURN R. Markov chains in random environments[J]. Ann Probab, 1980, 8(3): 908–916. [2] COGBURN R. The ergodic theory of Markov chains in random environments[J]. Probability Theory and Related Fields, 1984, 66(1): 109–128. [3] COGBURN R. On the central limit theorem for Markov chains in random environments[J]. Ann Probab, 1991, 19(2): 587–604. DOI:10.1214/aop/1176990442 [4] OREY S. Markov chains with stochastically stationary transition probabilities[J]. Ann Probab, 1991, 19(3): 907–928. DOI:10.1214/aop/1176990328 [5] 李应求. 双无限环境中马氏链的常返性与不变侧度[J]. 中国科学:A辑, 2001, 31(8): 702–707. LI Y Q. The recurrence and invariant measures for Markov chains in bi-infinite environments[J]. Science in China:Ser A, 2001, 31(8): 702–707. [6] 李应求. 双无限随机环境中马氏链的暂留性[J]. 数学物理学报:A辑, 2007, 27(2): 269–276. LI Y Q. Transience for Markov Chains in double infinite random environments[J]. Acta Mathematica Scientia:Ser A, 2007, 27(2): 269–276. [7] 李应求. 双无限随机环境中的常返马氏链[J]. 数学学报:A辑, 2007, 50(5): 1099–1100. LI Y Q. The recurrent Markov chains in bi-infinite random environments[J]. Acta Mathematica Sinica:Ser A, 2007, 50(5): 1099–1100. [8] 胡迪鹤. 随机环境中q-过程的存在唯一性[J]. 中国科学:A辑, 2004, 34(5): 625–640. HU D H. The existence and uniqueness of q-process Science in random environments[J]. Science in China:Ser A, 2004, 34(5): 625–640. [9] 胡迪鹤. 从P-m链到随军环境中的马氏链[J]. 数学年刊:A辑, 2004, 25(1): 65–78. HU D H. From P-m chains to Markov chains in random environments[J]. Chinese Annals of Mathematics:Ser A, 2004, 25(1): 65–78. [10] 王汉兴, 戴永隆. 马氏的Poisson极限律[J]. 数学学报, 1997, 40(2): 266–270. WANG H X, DAI Y L. Poisson limit law for Markov a chains in Markovian environments[J]. Acta Mathematica Sinica, 1997, 40(2): 266–270. [11] 方大凡. 马氏环境中马氏链的Shannon-McMillan-Breiman定理[J]. 应用概率统计, 2000, 16(3): 295–298. FANG D F. Shannon-McMillan-Breiman theorem for Markov chains in Markovian environments[J]. Chinese Journal of Applied Probability and Statistics, 2000, 16(3): 295–298. [12] 郭明乐. 马氏环境中的马氏链的强大数定律[J]. 应用数学, 2003, 16(4): 143–148. GUO M L. The strong law of large numbers for Markov chains in Markovian environments[J]. Mathematica Applicata, 2003, 16(4): 143–148. [13] 郭明乐. 随机环境中的马氏链的强大数定律[J]. 应用概率统计, 2004, 20(2): 154–160. GUO M L. The strong law of large numbers for Markov chains in random environments[J]. Chinese Journal of Applied Probability and Statistics, 2004, 20(2): 154–160. [14] 万成高. 马氏环境中马氏链的强大数定律[J]. 应用概率统计, 2003, 19(2): 155–160. WAN C G. On the strong law of large numbers for Markov chains in Markovian environments[J]. Chinese Journal of Applied Probability and Statistics, 2003, 19(2): 155–160. [15] 王伟刚. 一般随机环境中马氏链的强大数律[J]. 数学杂志, 2011, 31(3): 481–487. WANG W G. The strong law of large numbers for Markov chains in random environments[J]. Journal of Mathematics, 2011, 31(3): 481–487. [16] 胡迪鹤. 随机环境中的马尔可夫过程[M]. 北京: 高等教育出版社, 2011. HU D H. Markov Processes in Random Environments[M]. Beijing: Higher Education Press, 2011. [17] 朱成熹. 随机极限引论[M]. 天津: 南开大学出版社, 1987. ZHU C X. An Introduction to Random Limit[M]. Tianjin: Nankai University Press, 1987.