20世纪80年代初Cogburn R等人开始研究随机环境中马氏链的一般理论, 取得一系列深刻而丰富的成果^[1-3]. Orey ^[4]在Cogburn等人研究的基础对随机环境中马氏链进行了深入地研究, 并提出一系列的问题, 引起众多概率论学者的广泛关注.强大数定律是随机环境马氏链理论研究的热门课题之一, 已取得深入的结果, 然而马氏环境中马氏双链函数的强大数定律研究却很少.鉴于此, 本文引入了一类马氏双链函数, 主要研究其大数定律.本文利用将双链函数分段研究的方法, 得到了马氏环境中马氏双链函数强大数定律成立的一个充分条件, 并将该定律应用到马氏双链的研究中去, 得出马氏双链从一个状态到另一个状态转移概率的极限性质, 进而推广了马氏双链的极限性质.

本文沿用文[1-4]中的符号和术语, 设N表示整数集, N$_{+}$表示非负整数集, $(\Omega, \mathcal{F}, P)$是一概率空间, $(X, \mathcal{A})$和$(\Theta, \mathcal{B})$均为任意的可测空间, $\overrightarrow{\xi_{0}^{\infty}}=\{\xi_{n}:n\in N_{+}\}$和$\overrightarrow{X}=\{X_{n}:n\in N_{+}\}$分别是$(\Omega, \mathcal{F}, P)$上取值于$\Theta$和X的随机序列, $\{P(\theta):\theta\in\Theta\}$是$(X, \mathcal{A})$上的一族转移函数, 且假定对任意的$A\in\mathcal{A}, P(\cdot;\cdot, A)$关于$\mathcal{B}\times\mathcal{A}$可测的. $\{K_{n}(., .)\}$是$(\Theta, \mathcal{B})$上的一步转移概率函数族, 且假定对任意的$B\in\mathcal{B}, K_{n}(., B)$关于$\mathcal{B}$可测的, 对任意序列$\overrightarrow{\eta}=\{\eta_{n }, n\in N_{+}\}$, 记$\overrightarrow{\eta_{k}^{r }}=\{\eta_{n }, 0 \leq k\leq n \leq r \leq\infty\}.$

定义1 ^[5]  如果对任意的A$\in\mathcal{A}$, $n\in N_{+}$, 有

$ P(X_{0}\in A\mid\overrightarrow{\xi_{0}^{\infty}})=P(X_{0}\in A\mid \xi_{0}), \\ P(X_{n+1}\in A\mid\ \overrightarrow {X_{0}^{n}}, \overrightarrow {\xi_{0}^{\infty}})=P(\xi_{n};X_{n}, A), $

则称$\overrightarrow{X}$是单无限随机环境$\overrightarrow{\xi_{0}^{\infty}}$中的马氏链, $\overrightarrow{\xi_{0}^{\infty}}$为单无限随机环境序列.若$\overrightarrow{\xi_{0}^{\infty}}$是一马氏序列, 则称$\overrightarrow{X}$是单无限马氏环境中的马氏链.

引理1 ^[5]  设$\overrightarrow{X}$是单无限马氏环境$\overrightarrow{\xi_{0}^{\infty}}$中的马氏链, 则$\{(X_{n}, \xi_{n}), n\geq 0\}$是马氏双链.特别的, 若$\overrightarrow{\xi_{0}^{\infty}}$的一步转移函数为$K_{n}(\theta, B)$, 则$\{(X_{n}, \xi_{n}), n\geq 0\}$一步转移概率为

$Q_{n }(x, \theta;A\times B)=K_{n}(\theta, B)P(\theta;x, A); $

若$\overrightarrow{\xi_{0}^{\infty}}$是时齐的, 则$\{(X_{n}, \xi_{n}), n\geq 0\}$也是时齐的.

引理2 ^[6]   (克罗内克引理)设$\{x_{n}\}$是实数的一个序列, $\{\alpha_{n}, n>0\}$是$\lim\limits_{n \rightarrow\infty}\alpha_{n}=\infty $的一个正数列, 若$ \sum\limits_{n=1}^{\infty}\frac{x_{n}}{\alpha_{n}}<\infty, $则$\lim\limits_{n \rightarrow\infty} \frac{1}{\alpha_{n}}\sum\limits_{k=1}^{n}x_{k}=0.$

引理3 ^[6]  对任意的事件$\{E_{n}, n\in N_{+}\}$, 若$ \sum\limits_{n=0}^{\infty}P(E_{n}) <\infty, $则$P(E_{n}\;\;{\hbox{i.o.}})=0.$

本文恒设$\overrightarrow{X}$是单无限马氏环境中的马氏链.

定理1  设$\{(X_{n}, \xi_{n}), n\geq 0\}$是$(\Omega, \mathcal{F}, P)$上取值于X$\times \Theta $上的马氏双链, $\{F_{n}(X_{n}, \xi_{n}), n\geq 0\}$是$(X\times \Theta, \mathcal{A}\times \mathcal{B}) $上的可测函数列, 且$0<a_{n}\uparrow \infty$, 若

$\mathop \sum \limits_{n = 0}^\infty \textbf{E}\frac{|F_{n}(X_{n}, \xi_{n})|^{\beta}}{a_{n}|F_{n}(X_{n}, \xi_{n})|^{\beta-1}+a_{n}^{\beta}} ＜\infty (1\leq\beta\leq2), $

(2.1)

则对$\forall k\geq1$有

$\mathop \sum \limits_{n = 0}^\infty \frac{F_{n}(X_{n}, \xi_{n})-\textbf{E}(F_{n}(X_{n}, \xi_{n})|X_{n-k}, \xi_{n-k})}{a_{n}} <\infty\;\;\;\;{\hbox{a.s.}}, $

及

$\lim\limits_{n\rightarrow \infty}\frac{1}{a_{n}}\mathop \sum \limits_{n = 0}^\infty (F_{m}(X_{m}, \xi_{m})-\textbf{E}(F_{m}(X_{m}, \xi_{m})|X_{m-k}, \xi_{m-k})) =0\;\;\;\;{\hbox{a.s.}}, $

这里约定$\forall k\geq1, X_{-k}=0, \xi_{-k}=0$.

证  先考虑$ k=1$的情况.因为$1\leq\beta\leq2$, 所以当$|F_{n}(X_{n}, \xi_{n})|> a_{n}> 0$时, 有

$\frac{2|F_{n}(X_{n}, \xi_{n})|^{\beta}}{a_{n}|F_{n}(X_{n}, \xi_{n})|^{\beta-1}+a_{n}^{\beta}}>\frac{2|F_{n}(X_{n}, \xi_{n})|^{\beta}}{2|F_{n}(X_{n}, \xi_{n})|^{\beta}}=1\;\;, $

由(2.1) 式可知

$ \sum \limits_{n = 0}^\infty P(|F_{n}(X_{n}, \xi_{n})|>a_{n}) =\sum \limits_{n = 0}^\infty \textbf{E} I_{\{|F_{n}(X_{n}, \xi_{n})|>a_{n}\}}\\ \leq\sum \limits_{n = 0}^\infty \textbf{E}\frac{2|F_{n}(X_{n}, \xi_{n})|^{\beta}}{a_{n}|F_{n}(X_{n}, \xi_{n})|^{\beta-1}+a_{n}^{\beta}}I_{\{|F_{n}(X_{n}, \xi_{n})|>a_{n}\}}\\ \leq\sum \limits_{n = 0}^\infty \textbf{E}\frac{2|F_{n}(X_{n}, \xi_{n})|^{\beta}}{a_{n}|F_{n}(X_{n}, \xi_{n})|^{\beta-1}+a_{n}^{\beta}}<\infty. $

由引理3可知$P(|F_{n}(X_{n}, \xi_{n})|>a_{n} \;\; {\hbox{i.o.}})=0$, 即$P(\frac{|F_{n}(X_{n}, \xi_{n})|}{a_{n}}>1 \;\;{\hbox{i.o.}})=0$, 所以

$\sum \limits_{n = 0}^\infty \frac{F_{n}(X_{n}, \xi_{n})}{a_{n}}I_{\{|F_{n}(X_{n}, \xi_{n})|＞a_{n}\}}＜\infty \;\;\;\;{\hbox{a.s.}}, $

(2.2)

由(2.1) 式知

$ \textbf{E}\sum \limits_{n = 0}^\infty |\textbf{E}| \frac{F_{n}(X_{n}, \xi_{n})}{a_{n}}I_{\{|F_{n}(X_{n}, \xi_{n})|>a_{n}\}}|(X_{n-1}, \xi_{n-1})||\\ \leq \textbf{E}\sum \limits_{n = 0}^\infty |\textbf{E}\frac{|F_{n}(X_{n}, \xi_{n})|}{a_n}I_{\{|F_{n}(X_{n}, \xi_{n})|>a_{n}\}}|(X_{n-1}, \xi_{n-1})|\\ \leq \sum \limits_{n = 0}^\infty \textbf{E}(\frac{|F_{n}(X_{n}, \xi_{n})|}{a_{n}}\frac{2|F_{n}(X_{n}, \xi_{n})|^{\beta-1}}{a_{n}^{\beta-1}+|F_{n}(X_{n}, \xi_{n})|^{\beta-1}} I_{\{|F_{n}(X_{n}, \xi_{n})|>a_{n}\}}|(X_{n-1}, \xi_{n-1}))\\ \leq 2\sum \limits_{n = 0}^\infty \textbf{E}\frac{|F_{n}(X_{n}, \xi_{n})|^{\beta}}{a_{n}|F_{n}(X_{n}, \xi_{n})|^{\beta-1}+a_{n}^{\beta}} <\infty, $

所以

$\sum \limits_{n = 0}^\infty \textbf{E}( \frac{F_{n}(X_{n}, \xi_{n})}{a_n}I_{\{|F_{n}(X_{n}, \xi_{n})|＞a_{n}\}}|(X_{n-1}, \xi_{n-1}))＜\infty\;\;\;\;{\hbox{a.s.}}.$

(2.3)

记

$ Y_{n}=\frac{F_{n}(X_{n}, \xi_{n})}{a_n} I_{\{|F_{n}(X_{n}, \xi_{n})|\leq a_{n}\}}-\textbf{E} (\frac{F_{n}(X_{n}, \xi_{n})}{a_n} I_{\{|F_{n}(X_{n}, \xi_{n})|\leq a_{n}\}}|(X_{n-1}, \xi_{n-1})), $

$\mathfrak{B}_{n}=\sigma(\overrightarrow X_{0}^{n}, \overrightarrow \xi_{0}^{n} )$, 因为$\{(X_{n}, \xi_{n}), n\geq 0\}$是马氏双链, 所以$\{(Y_{n}, \mathfrak{B}_{n}), n\geq 0\}$为鞅差序列.由鞅差序列正交性以及$1\leq\beta\leq2$知

$ \textbf{E}|\mathop \sum \limits_{m = 0}^n Y_{m} |^{2} =\mathop \sum \limits_{m = 0}^n \textbf{E} Y_{m}^{2}\\ \leq4\mathop \sum \limits_{m = 0}^n \textbf{E} \frac{|F_{m}(X_{m}, \xi_{m})|^{2}}{a_{m}^{2}}I_{\{|F_{m}(X_{m}, \xi_{m})|\leq a_{m}\}}\\ \leq 4\mathop \sum \limits_{m = 0}^n \textbf{E} \frac{|F_{m}(X_{m}, \xi_{m})|^{\beta}}{a_{m}^{\beta}}I_{\{|F_{m}(X_{m}, \xi_{m})|\leq a_{m}\}}\\ \leq 4\mathop \sum \limits_{m = 0}^n \textbf{E} \frac{2|F_{m}(X_{m}, \xi_{m})|^{\beta}}{a_{m}^{\beta}+a_{m} |F_{m}(X_{m}, \xi_{m})|^{\beta-1}}I_{\{|F_{m}(X_{m}, \xi_{m})|\leq a_{m}\}}, $

由(2.1) 式可知

$\sup\limits_{n\geq0}\textbf{E}|\mathop \sum \limits_{m = 0}^n Y_{m} |^{2}<\infty, $

所以$\{(\sum\limits_{m=0}^{n}Y_{m}, \mathfrak{B}_{n}), n \geq 0\}$是$L^{2}$有界鞅, 从而

$\sum \limits_{n = 0}^\infty Y_{n}＜\infty \;\;\;\;{\hbox{a.s.}}, $

(2.4)

所以

$ \mathop \sum \limits_{n = 1}^\infty \frac{F_{n}(X_{n}, \xi_{n})-\textbf{E}(F_{n}(X_{n}, \xi_{n})|(X_{n-1}, \xi_{n-1}))}{a_{n}} \\ =\mathop \sum \limits_{n = 1}^\infty (\frac{F_{n}(X_{n}, \xi_{n})}{a_{n}} I_{\{|F_{n}(X_{n}, \xi_{n})|>a_{n}\}}) +\sum \limits_{n = 0}^\infty Y_{n} -\sum \limits_{n = 0}^\infty \textbf{E}( \frac{F_{n}(X_{n}, \xi_{n})}{a_{n}} I_{\{|F_{n}(X_{n}, \xi_{n})|>a_{n}\}})|(X_{n-1}, \xi_{n-1})), $

由(2.2), (2.3) 和(2.4) 式可知

$\mathop \sum \limits_{n = 1}^\infty \frac{F_{n}(X_{n}, \xi_{n})-\textbf{E}(F_{n}(X_{n}, \xi_{n})|(X_{n-1}, \xi_{n-1}))}{a_{n}} ＜\infty, \;\;\;\;{\hbox{a.s.}}. $

(2.5)

下面考虑$k>1$的情形.

由$\{(X_{n}, \xi_{n}), n\geq 0\}$的马氏性知, 对于任意的$m=1, 2, \cdots, k-1, \{(X_{nk+m}, \xi_{nk+m}), n\geq 0\}$也是马氏链, 由(2.1) 式显然有

$\sum \limits_{n = 0}^\infty \textbf{E}\frac{|F_{nk+m}(X_{nk+m}, \xi_{nk+m})|^{\beta}}{a_{nk+m}|F_{nk+m}(X_{nk+m}, \xi_{nk+m})|^{\beta-1}+a_{nk+m}^{\beta}} <\infty.$

由(2.5) 式知, 对任意的$m=1, 2, \cdots, k-1, $有

$\mathop \sum \limits_{n = 1}^\infty \frac{F_{nk+m}(X_{nk+m}, \xi_{nk+m})-\textbf{E}(F_{nk+m}(X_{nk+m}, \xi_{nk+m})|(X_{nk+m-k}, \xi_{nk+m-k}))}{a_{nk+m}} ＜\infty \;\;\;\;{\hbox{a.s.}}, $

(2.6)

从而由(2.6) 式知

$\sum\limits_{n = 1}^\infty {\frac{{{F_n}({X_n},{\xi _n}) - {\bf{E}}({F_n}({X_n},{\xi _n})|({X_{n - k}},{\xi _{n - k}}))}}{{{a_n}}}} \\ = \mathop \sum \limits_{n = 1}^\infty \sum\limits_{m=1}^{k-1}\frac{F_{nk+m}(X_{nk+m}, \xi_{nk+m})-\textbf{E}(F_{nk+m}(X_{nk+m}, \xi_{nk+m})|(X_{nk+m-k}, \xi_{nk+m-k}))}{a_{nk+m}}\nonumber\\ =\sum\limits_{m=1}^{k-1}\mathop \sum \limits_{n = 1}^\infty \frac{F_{nk+m}(X_{nk+m}, \xi_{nk+m})-\textbf{E}(F_{nk+m}(X_{nk+m}, \xi_{nk+m})|(X_{nk+m-k}, \xi_{nk+m-k}))}{a_{nk+m}}＜\infty. \;\;\;\;{\hbox{a.s.}}.$

(2.7)

由引理2和(2.7) 式知

$\mathop {\lim }\limits_{n \to \infty } \frac{1}{a_{n}}\mathop \sum \limits_{m = 1}^n F_{m}(X_{m}, \xi_{m})-\textbf{E}(F_{m}(X_{m}, \xi_{m})|(X_{m-k}, \xi_{m-k}))=0 {\hbox{a.s.}}.$

即定理1得证.

定理2  设$\overrightarrow{X}$是单无限马氏环境$\overrightarrow{\xi_{0}^{\infty}}$中的马氏链, $S_{n}(x, \theta)$表示序列$(X_{1}, \xi_{1}), (X_{2}, \xi_{2}), \cdots, \\(X_{n}, \xi_{n})$中$(x, \theta)$出现的次数, 则

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(x, \theta)}{n}-\frac{1}{n}\mathop \sum \limits_{m = 0}^n K_{m}(\xi_{m-1}, \theta)P(\xi_{m-1};X_{m-1}, x)=0\;\;\;\;{\hbox{a.s.}}. $

特别地, 当$\overrightarrow{\xi_{0}^{\infty}}$是时齐的, 有

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(x, \theta)}{n}=K (\xi_{0}, \theta)P(\xi_{0};X_{0}, x)\;\;\;\;{\hbox{a.s.}}. $

证  令$ F_{m}(X_{m}, \xi_{m})=\delta_{x}(X_{m})\delta_{\theta}(\xi_{m})$, 所以$S_{n}(x, \theta)=\sum\limits_{m=1}^{n}F_{m}(X_{m}, \xi_{m}), $则有

$\sum \limits_{n = 0}^\infty \textbf{E}\frac{|F_{n}(X_{n}, \xi_{n})|^{2}}{n|F_{n}(X_{n}, \xi_{n})|+n^{2}} \leq \sum \limits_{n = 0}^\infty \textbf{E}\frac{1}{n^{2}}<\infty, $

取$a_{n}=n, \beta=2 $, 由定理1可知

$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\mathop \sum \limits_{m = 0}^n (F_{m}(X_{m}, \xi_{m})-\textbf{E}(F_{m}(X_{m}, \xi_{m})|(X_{m-1}, \xi_{m-1})) =0 \;\;\;\;{\hbox{a.s.}}, $

(3.1)

又因为$\sum\limits_{m=0}^{n}F_{m}(X_{m}, \xi_{m})=S_{n}(x, \theta)+\delta_{x}(X_{0})\delta_{\theta}(\xi_{0}), $由引理1知

$ E(F_{m}(X_{m}, \xi_{m})|(X_{m-1}, \xi_{m-1}))=\textbf{E} (\delta_{x}(X_{m})\delta_{\theta}(\xi_{m})|(X_{m-1}, \xi_{m-1})\\ = Q_{m }(X_{m-1}, \xi_{m-1};x, \theta)=K_{m}(\xi_{m-1}, \theta)P(\xi_{m-1};X_{m-1}, x), $

将上式代入(3.1) 式得

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(x, \theta)}{n}-\frac{1}{n}\mathop \sum \limits_{m = 0}^n K_{m}(\xi_{m-1}, \theta) P(\xi_{m-1};X_{m-1}, x)=0\;\;\;\;{\hbox{a.s.}}.$

(3.2)

当$\overrightarrow{\xi_{0}^{\infty}}$是时齐的, 由引理1知$\{(X_{n}, \xi_{n}), n\geq 0\}$也是时齐的, 所以

$Q_{m }(X_{m-1}, \xi_{m-1};x, \theta) =Q(X_{0}, \xi_{0};x, \theta) =K(\xi_{0}, \theta)P(\xi_{0};X_{0}, x), $

从而由(3.2) 式得

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(x, \theta)}{n}=K (\xi_{0}, \theta)P(\xi_{0};X_{0}, x)\;\;\;\;{\hbox{a.s.}}. $

即定理2得证.

推论1  设$\overrightarrow{X}$是单无限马氏环境$\overrightarrow{\xi_{0}^{\infty}}$中的马氏链, $S_{n}(\theta)$表示序列$\xi_{1}, \xi_{2}, \cdots, \xi_{n}$中$\theta$出现的次数, 则

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(\theta)}{n}-\frac{1}{n}\mathop \sum \limits_{m = 0}^n K_{m}(\xi_{m-1}, \theta)=0\;\;\;\;{\hbox{a.s.}}.$

特别地, 当$\overrightarrow{\xi_{0}^{\infty}}$是时齐的, 有

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(\theta)}{n}=K (\xi_{0}, \theta)\;\;\;\;{\hbox{a.s.}}. $

证 令$ F_{m}(X_{m}, \xi_{m})=\delta_{\theta}(\xi_{m})$, 所以$S_{n}(\theta)+\delta_{\theta}(\xi_{0})=\sum\limits_{m=0}^{n}F_{m}(X_{m}, \xi_{m}), $由引理1知

$\textbf{E}(F_{m}(X_{m}, \xi_{m})|(X_{m-1}, \xi_{m-1}))=\textbf{E} (\delta_{\theta}(\xi_{m})|(X_{m-1}, \xi_{m-1}))=K_{m}(\xi_{m-1}, \theta).$

由定理2知$F_{n}(X_{n}, \xi_{n})$满足(2.1) 式, 由上述计算过程可得

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(\theta)}{n}-\frac{1}{n}\sum\limits_{m=0}^{n}K_{m}(\xi_{m-1}, \theta)=0\;\;\;\;{\hbox{a.s.}}; $

(3.3)

当$\overrightarrow{\xi_{0}^{\infty}}$是时齐的, 从而由(3.3) 式得

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(\theta)}{n}=K (\xi_{0}, \theta)\;\;\;\;{\hbox{a.s.}}. $

即推论1得证.

推论2  设$\overrightarrow{X}$是单无限马氏环境$\overrightarrow{\xi_{0}^{\infty}}$中的马氏链, $S_{n}(x)$表示序列$X_{1}, X_{2}, \cdots, X_{n}$中$x$出现的次数, 则

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(x)}{n}-\frac{1}{n}\mathop \sum \limits_{m = 0}^n P(\xi_{m-1};X_{m-1}, x)=0\;\;\;\;{\hbox{a.s.}}. $

特别地, 当$\overrightarrow{\xi_{0}^{\infty}}$是时齐的, 有

$\mathop {\lim }\limits_{n \to \infty } \frac{S_{n}(x)}{n}=P(\xi_{0};X_{0}, x)\;\;\;\;{\hbox{a.s.}}.$

证 令$ F_{m}(X_{m}, \xi_{m})=\delta_{x}(X_{m})$, 所以

$ S_{n}(x)+\delta_{x}(X_{0})=\mathop \sum \limits_{m = 0}^n F_{m}(X_{m}, \xi_{m})=\mathop \sum \limits_{m = 0}^n \delta_{x}(X_{m}), \\ \textbf{E}(F_{n}(X_{n}, \xi_{n})|(X_{n-1}, \xi_{n-1}))=\textbf{E} (\delta_{x}(X_{m})|(X_{n-1}, \xi_{n-1}))=P(\xi_{m-1};X_{m-1}, x), $

用类似于定理2和推论1的方法可得出推论2.