0 引言

马尔科夫链(以下简称经典马氏链)是随机过程中最重要的分支之一, 其理论已广泛应用于金融、保险、现代物理、分子生物学、随机服务系统等领域.基于经典马氏链理论研究, 考虑到经典马氏链中的转移函数会受外部随机因素的干扰, COGBURN^[1]增加了随机变量, 并引入了随机环境中的马尔科夫链模型(简称MCRE).自20世纪80年代以来, MCRE的理论研究得到了各国学者的广泛关注，并取得了丰富的成果.COGBURN在MCRE的状态分类与性质^[1-2]、中心极限定理^[3]等方面获得了丰富的研究成果.OREY^[4]对COGBURN等的工作进行了总结和评价, 同时得到类似于经典马氏链理论方面的很多成果并提出一些问题.李应求^[5-7]研究了MCRE的常返性.胡迪鹤^[8-9]给出了MCRE的构造.近年来, MCRE极限理论及其研究受到很多学者的重视, 王汉兴等^[10]研究了MCRE的Poisson极限律, 方大凡^[11]研究了MCRE的Shannon-McMillan-Breiman定理, 郭明乐^[12-13]、万成高^[14]、王伟刚^[15]分别研究了MCRE的强大数定律.胡迪鹤^[16]则对MCRE作了详细介绍与研究.本文主要讨论MCRE模型的推广, 通过引入多重随机环境中的马尔科夫链模型, 讨论m重随机环境中的马尔科夫链、k重随机环境中的马尔科夫链、MCRE、双链、2m维链的相互关系, 给出研究多重随机环境中马尔科夫链的一种方法, 以获得多重随机环境下马尔科夫链强大数定律成立的充分条件.推广了部分文献的结论.

1 m重MCRE模型及性质

设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 $={ξ_n, n=0, 1, …}和$\vec X $={X_n, n=0, 1, 2, …}是概率空间(Ω, $\mathscr{F}$, P)上分别取值于Θ和$\mathscr{X}$的2个随机序列, (Θ^m, $\mathscr{B}$^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 $_n^n+m-1=(ξ_n, ξ_n+1, …, ξ_n+m-1)和$\vec X $_n^n+m-1=(X_n, X_n+1, …, X_n+m-1)分别为(Θ^m, $\mathscr{B}$^m)与($\mathscr{X}$^m, $\mathscr{A}$^m)上的2个m维可测随机向量序列, Θ^m, $\mathscr{X}$^m中的元素分别用θ^m, x^m表示.

定义1    设p(·;·, ·):Θ^m×$\mathscr{X}$^m×$\mathscr{A} \mapsto$[0, 1], 且满足如下条件:

(ⅰ)对任意θ^m∈Θ^m及x^m∈$\mathscr{X}$^m, p(θ^m; x^m, ·)是$\mathscr{A}$上的概率测度.

(ⅱ)对任意θ^m∈Θ^m及A∈$\mathscr{A}$, p(θ^m; ·, A)关于$\mathscr{A}$^m可测.

(ⅲ)对任意x^m∈$\mathscr{X}$^m及A∈$\mathscr{A}$, p(·;x^m, 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.

定义2    设$\vec \xi $={ξ_n, n=0, 1, …}和$\vec X $={X_n, n=0, 1, 2, …}是概率空间(Ω, $\mathscr{F}$, P)上分别取值于Θ和$\mathscr{X}$的2个随机序列, p(·;·, ·)为Θ^m×$\mathscr{X}$^m×$\mathscr{A} \mapsto$[0, 1]上一个m重随机马尔科夫核, 若对任意的A∈$\mathscr{A}$, B∈$\mathscr{B}$, 下列2个条件成立:

$ 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)

则称$\vec X $为m重随机环境$\vec \xi $中的马尔科夫链.

注1    特别地, 当m=1时, $\vec X $为随机环境$\vec \xi $中的马尔科夫链^[2].

定理1    若$\vec X $为m重随机环境$\vec \xi $中的马尔科夫链, 且对任意A∈$\mathscr{A}$，p(·;·, A)作为(θ^k, x^k)的函数关于$\mathscr{B}$^k×$\mathscr{A}$^k可测, 则$\vec X $为k(k>m)重随机环境$\vec \xi $中的马尔科夫链.

证明    设p(·;·, ·)为一个m重随机马尔科夫核, 若记

$ \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} $

则可以诱导出一个k重随机马尔科夫核.设$\vec X $为m重随机环境$\vec \xi $中的马尔科夫链, 首先证明$\vec X $为m+1重随机环境$\vec \xi $中的马尔科夫链.由条件数学期望的性质知:

$ \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} $

从而由单调类定理易得

$ 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). $

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

注2    由注1及定理1知, m重随机环境中的马尔科夫链是随机环境中马尔科夫链模型的推广, 其直观想法为:系统或过程未来(n+m时刻)的演变规律只与最近一段时间内(n时刻到n+m-1时刻)系统或过程所处的状态和环境有关, 与过去(n时刻之前)无关.

下面给出该模型的2个应用背景.

例1(生物群体繁殖模型)    研究某种生物群体数量的演变规律, 以X_n表示第n个单位时刻生物群体的数量, ξ_n表示第n个单位时刻生物群体所处的外部环境, 若不考虑环境因素的变化(即假定ξ_n为常量), 且假定已知过去和现在群体数量, 下一个单位时间群体数量的演变规律只与现在有关, 即在X₀, X₁, …, X_n状态已知的条件下, X_n+1所处状态的转移规律只与X_n有关, 与X₀, X₁, …, X_n-1无关, 即为经典的马尔科夫链模型.更一般地, 考虑生物群体受生育和存活时间段及外部环境的随机变化影响, 在过去和现在群体数量和外部环境所处状态已知的条件下, 群体数量下一个单位时间的演变规律只与现在至过去一段时间内的群体数量和环境状态有关, 即在X₀, X₁, …, X_n+m-1, ξ₀, ξ₁, …, ξ_n+m-1所处状态已知的条件下, X_n+m只与X_n, X_n+1, …, X_n+m-1, ξ_n, ξ_n+1, …, ξ_n+m-1所处的状态有关, 该模型即为m重随机环境中的马尔科夫链.

例2(短线交易中的股票价格预测模型)    研究某种股票价格的预测模型, 若换手率较高, 则表明大部分投资人会在较短的时间内发生交易.将时间离散化, 以X_n, n=0, 1, 2, …表示在n时刻股票的价格, 以ξ_n, n=0, 1, 2, …表示在n时刻的外部环境(如经济政策、公司经营状况、市场环境等因素), 在不考虑环境因素变化(即假定ξ_n为常量)且假定在过去和现在股票价格已知的条件下, 股票价格下一个单位时间的演变规律只与现在有关，与过去无关, 即在已知X₀, X₁, …, X_n状态的条件下, X_n+1所处状态的转移规律只与X_n有关, 与X₀, X₁, …, X_n-1无关, 即为经典的马尔科夫链模型.该模型的假设过于理想化, 不符合股市的运行规律.一般情形下, 由于短期内大部分投资人会发生交易, 因此, 假设在已知过去和现在股票价格和外部环境所处状态的条件下, 股票价格下一个单位时刻的演变规律只与现在至过去一段时间内的股票价格和环境状态有关是合理的.即在已知X₀, X₁, …, X_n+m-1, ξ₀, ξ₁, …, ξ_n+m-1所处状态的条件下, X_n+m只与X_n, X_n+1, …, X_n+m-1, ξ_n, ξ_n+1, …, ξ_n+m-1所处的状态有关, 该模型即为m重随机环境中的马尔科夫链.

定理2    下列条件等价:

(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} $

定理3    若$\vec X $为m重随机环境$\vec \xi $中的马尔科夫链且$\vec \xi $为k重马尔科夫链, 则

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

(2) {W_n=(X_n, X_n+1, …, X_n+l-1, ξ_n, ξ_n+1, …, ξ_n+l-1), n≥0}为马尔科夫链.

证明    (1) 设$\vec X $为m重随机环境$\vec \xi $中的马尔科夫链, 由定理1知, $\vec X $为l重随机环境$\vec \xi $中的马尔科夫链.又因为$\vec \xi $={ξ_n, n≥0}为k重马尔科夫链, 故$\vec \xi $={ξ_n, n≥0}为l重马尔科夫链.设E为关于P的期望算子, 则由定理2知,

$ \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} $

故由单调类定理易得双链$\vec Y $={Y_n=(X_n, ξ_n), n≥0}为l重马尔科夫链.

(2) 令W_n=(Y_n, Y_n+1, …, Y_n+l-1), 则由(1) 知{Y_n=(X_n, ξ_n), 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\} $

为马尔科夫链.

推论1    若$\vec X $为m重随机环境$\vec \xi $中的马尔科夫链，且$\vec \xi $为m重马尔科夫链, 则

(1) 双链$\vec Y $={Y_n=(X_n, ξ_n), n≥0}为m重马尔科夫链.

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

推论2    若$\vec X $为m重随机环境$\vec \xi $中的马尔科夫链，且$\vec \xi $为m重时齐的马尔科夫链, 则

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

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

2 m重MCRE的强大数定律

本节总假定$\mathscr{X}$与Θ均为可列集, $\vec X $为m重随机环境$\vec \xi $中的马尔科夫链且$\vec \xi $是一个m重时齐的马尔科夫链, 则由定理3的推论2知，该条件保证了W_n=(Y_n, Y_n+1, …, Y_n+m-1)为时齐的马尔科夫链.本节总假设存在一个状态(x₀, θ₀, x₁, θ₁, …, x_m-1, θ_m-1), 使得条件:

$ \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)

成立, 即(x₀, θ₀, x₁, θ₁, …, x_m-1, θ_m-1)是W_n的一个常返状态.对每个k≥0, τ₀≡0, 定义一列马尔科夫时间为

$ \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} $

令

$ \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} $

再令l(n)=k, 当τ_k＜n≤τ_k+1时,

$ \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} $

则有

$ \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} $

引理1    (ⅰ){Z_k, k≥0}是概率空间(Ω, $\mathscr{F}$, P)上的独立随机变量列.

(ⅱ)若EZ_k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). $

证明    (ⅰ)由假设知{Z_k, k≥0}是取至多可列多个值的随机变量, 且对任意实数c_r(0≤r＜k)，有

$ \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} $

所以Y₀, Y₁, …, Y_k-1是$\mathscr{F}$_τk可测的, 令w=(x₀, θ₀, x₁, θ₁, …, x_m-1, θ_m-1), 则

$ \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}}}. $

从而由强马氏性质知

$ \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} $

利用数学归纳法易证引理1的(ⅰ)成立.

(ⅱ){l(n)＜m}={n≤τ_m}={τ_m＜n}^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} $

推论3    对任意实数a, b, {aZ_k+bσ_k, k≥0}与{aU_k+bσ_k, k≥0}均为(Ω, $\mathscr{F}$, P)上的独立随机变量列, 且{σ_k, k≥0}为(Ω, $\mathscr{F}$, P)上的独立同分布随机变量列.

条件I    存在非负且数学期望有限的随机变量Y, 使得对任意x>0, n∈N, 有

$ \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} $

引理2^[2]    若满足条件I, 则对任意n≥0, 有

(1) E|Z_n|+Eσ_n≤EU_n+Eσ_n≤EY＜∞;

(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→∞);

引理3^[17]    若满足条件I, 则

(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.

引理4    若满足条件I, 则

$ \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}}. $

证明    由l(n)的定义知:

$ \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} $

显然，上式最后一个等式第1项为0, 由引理1知第2项为0, 从而有

$ 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} } . $

于是有

$ \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} $

故由引理2知

$ \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}}. $

引理5    若满足条件I, 则

$ \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}}.} $

证明    由l(n)的定义知, l(n)→∞, a.s.(n→∞), 且$\frac{{l\left(n \right)}}{n}$＜1, 从而由引理3知

$ \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} $

即

$ \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}}.} $

定理4    若满足条件I, 则有下列强大数定律成立:

$ \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} $

证明    显然

$ \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} $

由引理2~5知, 上式每项均几乎处处收敛于0, 所以定理得证.

推论4    设g_i(·, ·)为满足条件I的二元函数, 则有下列强大数定律成立:

$ \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}}. $

证明    只需在定理3中令

$ {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). $

注3    推论4即为文献[13]的主要结果.