设$(\Omega, {\cal F})$是一个可测空间, $\mathbb{N}$是整数集, $\mathbb{N}_{+}$是非负整数集.令$\chi=\{0, 1, 2, ...\}$和$\Theta=\{0, 1, 2, ...\}$是两个可数状态空间, $\overrightarrow{\xi}=\{\xi_{n}, n\geq0\}$和$\overrightarrow{X}=\{X_{n}, n\geq0\}$是定义在可测空间$(\Omega, {\cal F})$上分别取值于$\Theta$和$\chi$的随机变量序列.设$p_{\theta}=\{p(\theta;x)\}, \theta\in\Theta$是含参数$\theta$的分布, $P_{\theta}=\{p(\theta;x, y), x, y\in\chi\}, \theta\in\Theta$是定义在$\chi^{2}$上含参数$\theta$的转移矩阵.对任意随机变量序列$\overrightarrow{\eta}=\{\eta_{n}, n=0, 1, ...\}$.记$\overrightarrow{\eta}_{k}^{r}=\{\eta_{n}, k\leq n\leq r\}$, $0\leq k\leq n\leq r\leq\infty$.设$\textbf{P}$是可测空间$(\Omega, {\cal F})$上的一个概率测度, 且在$\textbf{P}$下$\{(X_{n}, \xi_{n}), n\geq0\}$的有限维分布为

$ \begin{eqnarray*} \textbf{P}(\overrightarrow{X}_{0}^{n}=\overrightarrow{x}_{0}^{n}, \overrightarrow{\xi}_{0}^{n}=\overrightarrow{\theta}_{0}^{n}) &=& \textbf{P}(X_{0}=x_{0}, \xi_{0}=\theta_{0}, X_{1}=x_{1}, \xi_{1}=\theta_{1}, ..., X_{n}=x_{n}, \xi_{n}=\theta_{n}) \\ &=& p(\overrightarrow{x}_{0}^{n}, \overrightarrow{\theta}_{0}^{n})>0, \quad \forall (x_{i}, \theta_{i})\in (\chi \times\Theta). \end{eqnarray*} $

(1)

令

$ f_{n}(\omega)=-\frac{1}{n}\ln p(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n}), $

(2)

称$f_{n}(\omega)$为$\{(X_{n}, \xi_{n}), n\geq0\}$的熵密度, 其中$\ln$是以e为底的自然对数.

定义1 设$\overrightarrow{\xi}=\{\xi_{n}, n\geq0\}$和$\overrightarrow{X}=\{X_{n}, n\geq0\}$是定义在可测空间$(\Omega, {\cal F})$上分别取值于$\Theta$和$\chi$的随机变量序列.设$\textbf{Q}$为可测空间$(\Omega, {\cal F})$上的另一个概率测度, 若对任意$x, y\in\chi, n\in\mathbb{N_{+}}$, 有

$ \begin{eqnarray*} &&\textbf{Q}(X_{0}=x_{0}|\overrightarrow{\xi})=\textbf{Q}(X_{0}=x_{0}|\xi_{0})=q_{0}(\xi_{0};x_{0})\quad \text{a.e.}, \\ &&\textbf{Q}(X_{n+1}=y|\overrightarrow{X_{0}^{n}}, \overrightarrow{\xi})=p(\xi_{n};X_{n}, y)\quad \text {a.e.}, \end{eqnarray*} $

则称$\overrightarrow{X}$在概率测度Q下为单无限随机环境$\overrightarrow{\xi}$下的马氏链.特别地, 若$\overrightarrow{\xi}$是马氏链, 则称$\overrightarrow{X}$在概率测度Q下为单无限马氏环境$\overrightarrow{\xi}$下的马氏链.

易知, 若$\overrightarrow{X}$在概率测度Q下为单无限马氏环境$\overrightarrow{\xi}$下的马氏链, 则$\{(X_{n}, \xi_{n}), n\geq0\}$在概率测度$\textbf{Q}$下是马氏双链^[1].特别地, 若$\{\xi_{n}, n\geq0\}$是初始分布为$p^{\prime}(\theta_{0})$, 转移概率为$K(\theta, \alpha)$的马氏链, 则在概率测度$\textbf{Q}$下, $\{(X_{n}, \xi_{n}), n\geq0\}$是一个具有初始分布

$ q(x_{0}, \theta_{0})=p^{\prime}(\theta_{0})p(\theta_{0};x_{0}) $

(3)

和转移矩阵

$ P(x, \theta;y, \alpha)=K(\theta, \alpha)p(\theta;x, y) $

(4)

的马氏双链.设$\{(X_{n}, \xi_{n}), n\geq0\}$在概率测度Q下的有限维分布为

$ \begin{eqnarray*} \textbf{Q}(\overrightarrow{X}_{0}^{n}=\overrightarrow{x}_{0}^{n}, \overrightarrow{\xi}_{0}^{n}=\overrightarrow{\theta}_{0}^{n}) & = & \textbf{Q}(X_{0}=x_{0}, \xi_{0}=\theta_{0}, X_{1}=x_{1}, \xi_{1}=\theta_{1}, ..., X_{n}=x_{n}, \xi_{n}=\theta_{n}) \\ & = & q(\overrightarrow{x}_{0}^{n}, \overrightarrow{\theta}_{0}^{n})>0, \quad \forall (x_{i}, \theta_{i})\in (\chi \times \Theta). \end{eqnarray*} $

(5)

易知, 若$\{(X_{n}, \xi_{n}), n\geq0\}$在概率测度Q下为马氏环境下马氏链, 则

$ q(\overrightarrow{x}_{0}^{n}, \overrightarrow{\theta}_{0}^{n})=q(x_{0}, \theta_{0})\prod\limits_{i=1}^{n}P(x_{i-1}, \theta_{i-1};x_{i}, \theta_{i}), $

(6)

且

$ -\frac{1}{n}\ln q(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})=-\frac{1}{n}[\ln q(X_{0}, \xi_{0})+\sum\limits_{i=1}^{n}\ln P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})]. $

(7)

设概率测度P和Q为可测空间$(\Omega, {\cal F})$上的概率测度, 令

$ h(\textbf{P}|\textbf{Q})=\limsup\limits_{n \to \infty}\frac{1}{n}\ln\frac{p(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}{q(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}, $

(8)

称$h(\textbf{P}|\textbf{Q})$为P关于Q的渐近对数似然比.特别地, 若$\{(X_{n}, \xi_{n}), n\geq0\}$在概率测度Q下是单无限马氏环境下的马氏链, 则

$ h(\textbf{P}|\textbf{Q})=\limsup\limits_{n \to \infty}\frac{1}{n}\ln\Big[\frac{p(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}{q(X_{0}, \xi_{0})\prod\limits_{i=1}^{n}P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}\Big]. $

(9)

随机环境中的马氏链的研究已有相当长的历史, Nawrotzkill^{[2, 3]}建立了随机环境中的一般理论. Cogburn^{[1, 4, 5]}构造了Hopf-链, 利用Hopf-链理论深入研究了平稳环境中马氏链的遍历理论, 中心极限定理, 直接收敛和转移函数的周期性关系以及不变概率测度的存在性. Hu^[6-8]对连续时间参数的随机环境中的马氏过程的存在性, 等价性, $q$-过程的存在唯一性进行了研究.李应求等^[9-11]利用鞅差理论来研究随机环境中的马氏链, 在假设马氏双链遍历的条件下, 得到了马氏环境中马氏链的强大数定律成立的充分条件以及马氏环境中若干强极限定理.石志岩等^[12-14]研究了随机环境下树指标马氏链的定义及其存在性, 以及马氏环境下Cayley树指标马氏链的Shannon-McMillan定理.

强偏差定理(亦称小偏差定理)是由不等式表示的一类强极限定理, 它是强极限定理的推广.刘文和杨卫国^[15]研究了马氏逼近和任意随机变量序列的一类小偏差定理; 杨卫国^[16]研究了任意$N$值随机变量序列关于$m$阶非齐次马氏链的一类小偏差定理; 彭维才^[17]研究了关于齐次树上随机场的一类强偏差定理; 石志岩^[18]研究了树指标随机过程的一类强偏差定理; 石志岩和季金莉等^[19]研究了可列齐次马氏链的一类强偏差定理.

本文首先给出渐近对数似然比的概念, 通过构造非负鞅的方法, 建立可列状态齐次马氏链的强偏差定理.最后, 我们得到了单无限马氏环境下可列齐次马氏链的强大数定律及Shannon-McMillan定理.

引理1

$ h(\textbf{P}|\textbf{Q})\geq0, \quad \textbf{P}-\text{a.e.}. $

(10)

证 该引理的详细证明与文献[20]的引理1类似, 故此处省略.

引理2 设P 和Q是定义在可测空间$(\Omega, {\cal F})$上的两个概率测度, 且$\{(X_{n}, \xi_{n}), n\geq0\}$在概率测度Q下为定义$1$中定义的单无限马氏环境下的马氏链. $f(x, \theta;y, \alpha)$是$(\chi\times\Theta)^{2}$上的实函数, $\lambda$为一个实数.令

$ t_{n}(\lambda, \omega)=\frac{e^{\lambda \sum\limits_{i=1}^{n}f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}}{\prod\limits_{i=1}^{n}E_{\textbf{Q}}[e^{\lambda f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}|X_{i-1}, \xi_{i-1}]}\frac{q(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}{p(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}, \quad n=1, 2, ..., $

(11)

其中$E_{\textbf{Q}}$表示在概率测度Q下的期望, 则$\{t_{n}(\lambda, \omega), {\cal F}_{n}, n\geq1 \}$是概率测度P 下的非负鞅.

证 令${\cal F}_{n}=\sigma\{\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n}\}$, 则

$ \begin{eqnarray*} &\, \, & E_{\textbf{P}}[t_{n}(\lambda, \omega)|{\cal F}_{n-1}]\\ & = & E_{\textbf{P}}\Big[\frac{e^{\lambda \sum\limits_{i=1}^{n}f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}}{\prod\limits_{i=1}^{n}E_{\textbf{Q}}[e^{\lambda f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}|X_{i-1}, \xi_{i-1}]}\frac{q(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}{p(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})} |\overrightarrow{X}_{0}^{n-1}, \overrightarrow{\xi}_{0}^{n-1}\Big]\\ & = & t_{n-1}(\lambda, \omega)\cdot E_{\textbf{P}}\Big[\frac{e^{\lambda f(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}}{E_{\textbf{Q}}[e^{\lambda f(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}|X_{n-1}, \xi_{n-1}]}\frac{P(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}{p(X_{n}, \xi_{n}|\overrightarrow{X}_{0}^{n-1}, \overrightarrow{\xi}_{0}^{n-1})}|\overrightarrow{X}_{0}^{n-1}, \overrightarrow{\xi}_{0}^{n-1}\Big]\\ &=&t_{n-1}(\lambda, \omega)\cdot \frac{1}{E_{\textbf{Q}}[e^{\lambda f(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}|X_{n-1}, \xi_{n-1}]}\\ &&\cdot E_{\textbf{P}}\Big[e^{\lambda f(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}\cdot\frac{P(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}{p(X_{n}, \xi_{n}|\overrightarrow{X}_{0}^{n-1}, \overrightarrow{\xi}_{0}^{n-1})}|\overrightarrow{X}_{0}^{n-1}, \overrightarrow{\xi}_{0}^{n-1}\Big]\\ & = & t_{n-1}(\lambda, \omega)\cdot \frac{1}{E_{\textbf{Q}}[e^{\lambda f(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}|X_{n-1}, \xi_{n-1}]} \\ &&\cdot \sum\limits_{(y, \alpha)\in(\chi, \Theta)}e^{\lambda f(X_{n-1}, \xi_{n-1};y, \alpha)} \frac{P(X_{n-1}, \xi_{n-1};y, \alpha)}{p(X_{n}=y, \xi_{n}=\alpha|\overrightarrow{X}_{0}^{n-1}, \overrightarrow{\xi}_{0}^{n-1})}p(X_{n}=y, \xi_{n}=\alpha|\overrightarrow{X}_{0}^{n-1}, \overrightarrow{\xi}_{0}^{n-1})\\ & = & t_{n-1}(\lambda, \omega)\cdot \frac{\sum\limits_{(y, \alpha)\in(\chi, \Theta)}e^{\lambda f(X_{n-1}, \xi_{n-1};y, \alpha)}P(X_{n-1}, \xi_{n-1};y, \alpha)}{E_{\textbf{Q}}[e^{\lambda f(X_{n-1}, \xi_{n-1};X_{n}, \xi_{n})}|X_{n-1}, \xi_{n-1}]} & = & t_{n-1}(\lambda, \omega)\qquad \textbf{P}-\text{a.e.}. \end{eqnarray*} $

(12)

因此$\{t_{n}(\lambda, \omega), {\cal F}_{n}, n\geq1\}$是概率测度P 下的非负鞅.

定理1 设P 和Q是定义在可测空间$(\Omega, {\cal F})$上的两个概率测度, $\{(X_{n}, \xi_{n}), n\geq0\}$在概率测度Q下为取值于可数状态空间$(\chi\times\Theta)$的单无限马氏环境下的马氏链, $f(x, \theta;y, \alpha)$是定义于$(\chi\times\Theta)^{2}$上的实函数.令$c\geq0$,

$ D(c)=\{\omega, h(\textbf{P}|\textbf{Q})\leq c\}. $

(13)

假设存在实数$\alpha>0$, 对任意正整数$k$, 有

$ E_{\textbf{Q}}[e^{\alpha|f(X_{k-1}, \xi_{k-1};X_{k}, \xi_{k})|}]<\infty, $

(14)

且任意$(x, \theta)\in(\chi\times\Theta)$, 有

$ B_{\alpha}(x, \theta)=E_{\textbf{Q}}[f^{2}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})e^{\alpha|f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}|X_{i-1}=x, \xi_{i-1}=\theta]\leq \tau. $

(15)

其中$E_{\textbf{Q}}$表示在概率测度Q下的期望.则有

$ \begin{eqnarray*} &\, \, & \limsup\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}} [f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\leq& \sqrt{2c\tau}, \quad \textbf{P}-\text{a.e.}, \ \ \omega\in D(c). \end{eqnarray*} $

(16)

$ \begin{eqnarray*} &\, \, & \liminf\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}} [f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\geq& -\sqrt{2c\tau}, \quad \textbf{P}-\text{a.e.}, \ \ \ \omega\in D(c). \end{eqnarray*} $

(17)

特别地

$ \begin{eqnarray*} &\, \, & \lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}} [f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}{}\nonumber\\ & = &0, \quad \textbf{P}-\text{a.e.}, \ \ \ \omega\in D(0). \end{eqnarray*} $

(18)

证 由引理2知, $\{t_{n}(\lambda, \omega), {\cal F}_{n}, n\geq1\}$是在概率测度$\textbf{P}$下是非负鞅, 由Doob鞅收敛定理^[21]知, 存在一个有限非负随机变量$t_{\infty}(\lambda, \omega)$使得$ \lim\limits_{n\rightarrow\infty}t_{n}(\lambda, \omega)=t_{\infty}(\lambda, \omega), \quad \textbf{P}-\text{a.e.}, $故

$ \limsup\limits_{n\rightarrow\infty}\frac{1}{n}\ln t_{n}(\lambda, \omega)\leq0, \quad \textbf{P}-\text{a.e.}. $

(19)

由(10)和(19)式有

$ \begin{eqnarray*} &\, \, & \limsup\limits_{n\rightarrow\infty}\frac{1}{n}\Big[\sum\limits_{i=1}^{n}\{\lambda f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-\ln E_{\textbf{Q}}[e^{\lambda f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}|X_{i-1}, \xi_{i-1}]\}\\ &-&\ln\frac{p(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}{q(\overrightarrow{X}_{0}^{n}, \overrightarrow{\xi}_{0}^{n})}\Big]\leq0. \quad \textbf{P}-\text{a.e.}. \end{eqnarray*} $

(20)

由(9), (13), (19)和(20)式有

$ \begin{eqnarray*} &\, \, & \limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{\lambda f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-\ln E_{\textbf{Q}}[e^{\lambda f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}|X_{i-1}, \xi_{i-1}]\}\leq c\\ &\, \, & \textbf{P}-\text{a.e.}, \ \ \ \omega\in D(c). \end{eqnarray*} $

(21)

当$0<|\lambda|\leq\alpha$, 由(11), (21)以及不等式$\ln x\leq x-1(x>0)$和$e^{x}-x-1\leq(x^{2}/2)e^{|x|}$可得

$ \begin{eqnarray*} &\, \, &\limsup\limits_{n\rightarrow\infty}\frac{\lambda}{n}\sum\limits_{i=1}^{n}\{f-E_{\textbf{Q}}[f|X_{i-1}, \xi_{i-1}]\}\\ &\leq&\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{\ln E_{\textbf{Q}}[e^{\lambda f}|X_{i-1}, \xi_{i-1}]-E_{\textbf{Q}}[\lambda f|X_{i-1}, \xi_{i-1}]\}+c\\ &\leq&\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{E_{\textbf{Q}}[e^{\lambda f}|X_{i-1}, \xi_{i-1}]-1-E_{\textbf{Q}}[\lambda f|X_{i-1}, \xi_{i-1}]\}+c\\ &\leq&\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[(e^{\lambda f}-1-\lambda f)|X_{i-1}, \xi_{i-1}]+c\\ &\leq&\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[(\lambda^{2}/2)f^{2}e^{|\lambda f|}|X_{i-1}, \xi_{i-1}]+c\\ &=&\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}(\lambda^{2}/2)E_{\textbf{Q}}[f^{2}e^{\alpha|f|}e^{(|\lambda|-\alpha)|f|}|X_{i-1}, \xi_{i-1}]+c\\ &\leq&\frac{\lambda^{2}}{2}\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[f^{2}e^{\alpha|f|}|X_{i-1}, \xi_{i-1}]+c\\ &\leq&\frac{\lambda^{2}\tau}{2}+c, \quad \textbf{P}-\text{a.e.}, \ \ \ \omega\in D(c), \end{eqnarray*} $

(22)

其中$f\triangleq f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})$.当$0<\lambda\leq\alpha$, 由(22)式得

$ \begin{eqnarray*} &\, \, &\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\leq& \frac{\lambda}{2}\tau+\frac{c}{\lambda}, \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(23)

令$g(\lambda)=\frac{\lambda}{2}\tau+\frac{c}{\lambda}$, 易知, 当$\lambda=\sqrt{\frac{2c}{\tau}}$时, $g(\lambda)$取最小值$\sqrt{2c\tau}$, 在不等式(23)中令$\lambda=\sqrt{\frac{2c}{\tau}}$, 则有

$ \begin{eqnarray*} &\, \, &\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\leq& \sqrt{2c\tau}, \quad \textbf{P}-a.e., \omega\in D(c). \end{eqnarray*} $

(24)

当$c=0$时, 令$\lambda\rightarrow 0^{+}$, 由(23)式有

$ \begin{eqnarray*} &\, \, &\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\leq& 0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(25)

当$-\alpha\leq\lambda<0$, 由(22)式得

$ \begin{eqnarray*} &\, \, &\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\geq& \frac{\lambda}{2}\tau+\frac{c}{\lambda}, \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(26)

令$g(\lambda)=\frac{\lambda}{2}\tau+\frac{c}{\lambda}$, 易知, 当$\lambda=-\sqrt{\frac{2c}{\tau}}$时, $g(\lambda)$取最大值$-\sqrt{2c\tau}$, 在不等式(26)中令$\lambda=-\sqrt{\frac{2c}{\tau}}$, 则有

$ \begin{eqnarray*} &\, \, &\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\geq& -\sqrt{2c\tau}, \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(27)

当$c=0$时, 令$\lambda\rightarrow 0^{-}$, 由(26)式有

$ \begin{eqnarray*} &\, \, &\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\geq& 0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(28)

由(25)和(28)式知(18)式成立.

推论1 在定理$1$的条件下, 若把条件$(15)$式改为

$ B_{\alpha}(x, \theta)=E_{\textbf{Q}}[e^{\alpha|f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}|X_{i-1}=x, \xi_{i-1}=\theta]\leq \tau. $

(29)

则有

$ \begin{eqnarray*} &\, \, &\limsup\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}} [f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\leq& \inf\limits_{\lambda\in(0, \alpha]}g(\lambda), \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(30)

$ \begin{eqnarray*} &\, \, &\liminf\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}} [f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\geq&\sup\limits_{\lambda\in[-\alpha, 0)}\{-g(\lambda)\}, \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(31)

其中$g(\lambda)=\frac{2\tau\lambda e^{-2}}{(\alpha-\lambda)^{2}}+\frac{c}{\lambda}$.特别地,

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}} [f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ & = &0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(32)

证 易知$\max\{x^{2}e^{-hx}, x\geq0\}=\frac{4e^{-2}}{h^{2}}, h>0$.由$(22)$和$(29)$式知

$ \begin{eqnarray*} &&\limsup\limits_{n\rightarrow\infty}\frac{\lambda}{n}\sum\limits_{i=1}^{n}\{f-E_{\textbf{Q}}[f|X_{i-1}, \xi_{i-1}]\}\\ &\leq&\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}(\lambda^{2}/2)E_{\textbf{Q}}[e^{\alpha|f|}f^{2}e^{(|\lambda|-\alpha)|f|}|X_{i-1}, \xi_{i-1}]+c\\ &\leq&\frac{\lambda^{2}}{2}\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[e^{\alpha|f|}\frac{4e^{-2}}{(\alpha-|\lambda|)^{2}}|X_{i-1}, \xi_{i-1}]+c\\ &\leq&\frac{2\tau \lambda^{2}e^{-2}}{(\alpha-|\lambda|)^{2}}+c, \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(33)

其中$f\triangleq f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})$.当$0<\lambda\leq\alpha$, 由(33)式知

$ \begin{eqnarray*} &\, \, &\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\leq& \frac{2\tau\lambda e^{-2}}{(\alpha-\lambda)^{2}}+\frac{c}{\lambda}, \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(34)

当$c=0$时, 令$\lambda\rightarrow 0^{+}$, 由(34)式知

$ \begin{eqnarray*} &\, \, &\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\leq& 0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(35)

当$-\alpha\leq\lambda<0$, 由(33)式知

$ \begin{eqnarray*} &\, \, &\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\geq& -\bigg[\frac{2\tau\lambda e^{-2}}{(\alpha-\lambda)^{2}}+\frac{c}{\lambda}\bigg], \quad \textbf{P}-\text{a.e.}, \omega\in D(c). \end{eqnarray*} $

(36)

当$c=0$时, 令$\lambda\rightarrow 0^{-}$, 由(36)式知

$ \begin{eqnarray*} &\, \, &\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]\}\\ &\geq& 0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(37)

由(34)和(36)式知(30), (31)式成立, 由(35)和(37)式知(32)式成立.

定理2 在定理$1$的条件下, 令转移矩阵P是强遍历的, $\pi$是由P决定的唯一平稳分布, 若对任意$(x, \theta), (y, \alpha)\in(\chi\times\Theta)$, 有

$ \sup\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)<\infty. $

(38)

$ C_{\alpha}(x, \theta)=E_{\textbf{Q}}[f^{2}(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})e^{\alpha|f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|}|X_{i-1}=x, \xi_{i-1}=\theta]\leq \tau. $

(39)

对任意正整数$k$, 有

$ E_{\textbf{Q}}[e^{\alpha|f(X_{k}, \xi_{k};X_{k+1}, \xi_{k+1})|}]<\infty, $

(40)

其中$E_{\textbf{Q}}$表示在概率测度Q下的期望, 则有

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})=\sum\limits_{(x, \theta)\in(\chi, \Theta)}\ \pi(x, \theta)\sum\limits_{(y, \alpha)\in(\chi, \Theta)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)\\ &\, \, &\textbf{P}-\text{a.e.}, \ \ \ \omega\in D(0). \end{eqnarray*} $

(41)

证 由定理1知

$ \begin{eqnarray*} &\, \, &\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]}\}=0\\ &\, \, &\textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(42)

由(38)式知, 对任意$(x, \theta)\in(\chi, \Theta)$, 有$ \sum\limits_{(y, \alpha)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)<\infty, $故对任意$i\geq1$, 有$E_{\textbf{Q}}[f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]<\infty$, 因此可得

$ \lim\limits_{n\to\infty}\frac{E_{\textbf{Q}}[f(X_{n}, \xi_{n};X_{n+1}, \xi_{n+1})|X_{n}, \xi_{n}]}{n}=0, $

(43)

$ \lim\limits_{n\to\infty}\frac{E_{\textbf{Q}}[f(X_{0}, \xi_{0};X_{1}, \xi_{1})|X_{0}, \xi_{0}]}{n}=0. $

(44)

由(42), (43)和(44)式可得

$ \begin{eqnarray*} &\, \, &\lim\limits_{n\to\infty}\{\frac{1}{n}\sum\limits_{i=1}^{n}f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i}, \xi_{i}]\}=0, \\ &\, \, &\textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(45)

令$f_{1}(x, \theta;y, \alpha)=E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i}=y, \xi_{i}=\alpha]$.易知$e^{\alpha|x|}$是凸函数, 利用条件期望的Jensen不等式有

$ \begin{eqnarray*} &&E_{\textbf{Q}}[e^{\alpha|f_{1}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}] = E_{\textbf{Q}}[e^{\alpha|E_{Q}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i}, \xi_{i}]|}]\\ &\leq&E_{\textbf{Q}}[E_{Q}[e^{\alpha|f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|}|X_{i}, \xi_{i}]] = E_{\textbf{Q}}[e^{\alpha|f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|}]<\infty. \end{eqnarray*} $

易知$g(x)=x^{2}e^{\alpha|x|}$也是一个凸函数, 由(39)式和条件期望的Jensen不等式, 有

$ \begin{eqnarray*} &\, \, &\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[f_{1}^{2}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})e^{\alpha|f_{1}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}|X_{i-1}, \xi_{i-1}]\\ &=&\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[g(f_{1}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i}))|X_{i-1}, \xi_{i-1}]\\ &=&\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[g(E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i}, \xi_{i}]|X_{i-1}, \xi_{i-1})]\\ &\leq&\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[E_{\textbf{Q}}[g(f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1}))|X_{i}, \xi_{i}]|X_{i-1}, \xi_{i-1}]\\ &=&\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[g(f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1}))|X_{i-1}, \xi_{i-1}]\\ &=&\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[f^{2}(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})e^{\alpha|f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|}|X_{i-1}, \xi_{i-1}]\\ &\leq&\tau. \end{eqnarray*} $

因此$f_{1}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})$满足定理1的条件(14)和(15), 故由定理1可得

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{{f_{1}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{Q} [f_{1}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}]}\}\\ &=&0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(46)

注意到

$ \begin{eqnarray*} E_{\textbf{Q}}[f_{1}(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|X_{i-1}, \xi_{i-1}] &=&E_{\textbf{Q}}[E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i}, \xi_{i}]|X_{i-1}, \xi_{i-1}]\\ &=&E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i-1}, \xi_{i-1}], \end{eqnarray*} $

(47)

由(46)和(47)式可得

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i}, \xi_{i}]-E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i-1}, \xi_{i-1}]\}\\ &=&0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(48)

由(43)和(44)式, 有

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{E_{\textbf{Q}}[f(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|X_{i}, \xi_{i}]-E_{\textbf{Q}}[f(X_{i+1}, \xi_{i+1};X_{i+2}, \xi_{i+2})|X_{i}, \xi_{i}]\}\\ &=&0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(49)

由(45)和(49)式, 有

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i+1}, \xi_{i+1};X_{i+2}, \xi_{i+2})|X_{i}, \xi_{i}]\}\\ &=&0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(50)

由归纳可知, 对任意正整数$h$, 有

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\{f(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})-E_{\textbf{Q}}[f(X_{i+h}, \xi_{i+h};X_{i+h+1}, \xi_{i+h+1})|X_{i}, \xi_{i}]\}\\ &=&0, \quad \textbf{P}-\text{a.e.}, \omega\in D(0). \end{eqnarray*} $

(51)

因为$P$是强遍历的, 且$\pi$是由$P$决定的唯一平稳分布, 则

$ \begin{eqnarray*} &&\bigg |\frac{1}{n}\sum\limits_{i=1}^{n}E_{\textbf{Q}}[f(X_{i+h}, \xi_{i+h};X_{i+h+1}, \xi_{i+h+1})|X_{i}, \xi_{i}] -\sum\limits_{(x, \theta)}\pi(x, \theta)\sum\limits_{(y, \alpha)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)\bigg |\\ &=&\bigg|\frac{1}{n}\sum\limits_{i=1}^{n}\sum\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)P^{(h)}(X_{i}, \xi_{i};x, \theta)\\ &&-\sum\limits_{(x, \theta)}\pi(x, \theta)\sum\limits_{(y, \alpha)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)\bigg|\\ &=&\bigg|\frac{1}{n}\sum\limits_{i=1}^{n}\sum\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}\sum\limits_{l}\delta_{l}(X_{i})\sum\limits_{k}\delta_{k}(\xi_{i})f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)P^{(h)}(l, k;x, \theta)\\ &&-\sum\limits_{(x, \theta)}\pi(x, \theta)\sum\limits_{(y, \alpha)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)\bigg|\\ &=&\bigg|\frac{1}{n}\sum\limits_{l}\delta_{l}(X_{i})\sum\limits_{k}\delta_{k}(\xi_{i})\sum\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}f(x, \theta;y, \alpha)P(x, \theta;y, \alpha)[P^{(h)}(l, k;x, \theta)-\pi(x, \theta)]\bigg|\\ &\leq&\sup\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}\big|f(x, \theta;y, \alpha)\big|\cdot P(x, \theta;y, \alpha)\cdot\sup\limits_{(l, k)}\sum\limits_{(x, \theta)}\big|P^{(h)}(l, k;x, \theta)-\pi(x, \theta)\big|\\ &\rightarrow&0\quad(h\rightarrow\infty). \end{eqnarray*} $

(52)

由(51)和(52)式可知, (41)式成立.

$f_{n}(\omega)$在某种意义下收敛于常数($L_{1}$收敛, 依概率收敛, $\text{a.e.}$收敛), 在信息论中称为Shannon-McMillan定理或熵定理或称为信源的渐近等分性(AEP). Shannon^[22]首先研究了有限字母集上的平稳遍历信源依概率收敛的渐近等分性; McMillan^[23]和Breiman^[24]分别证明有限字母集上的平稳遍历信源$L_{1}$收敛和a.e.收敛的渐近等分性; 钟开莱^[25]研究了字母集为可列的情况; 刘文和杨卫国^{[26, 27]}给出了一类非齐次马氏信源的渐近等分性以及$m$阶非齐次马氏信源的渐近等分性.本节我们将研究单无限马氏环境下可列齐次马氏链的Shannon-McMillan定理.

定理3 设$\{(X_{n}, \xi_{n}), n\geq0\}$是取值于可数状态空间$(\chi\times\Theta)$的随机变量序列.设$P$是强遍历的, 且$\pi$是由P决定的唯一平稳分布.若

$ \sup\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}P^{\frac{1}{2}}(x, \theta;y, \alpha)\ln^{2}P(x, \theta;y, \alpha)<\infty, $

(53)

$ \sup\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}P(x, \theta;y, \alpha)|\ln P(x, \theta;y, \alpha)|<\infty, $

(54)

则

$ \lim\limits_{n \to \infty}f_{n}(\omega)=-\sum\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}\pi(x, \theta)P(x, \theta;y, \alpha)\ln P(x, \theta;y, \alpha), \quad \textbf{P}-\text{a.e.}, \omega\in D(0). $

(55)

证 由(53)式可得$\sup\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}P^{\frac{1}{2}}(x, \theta;y, \alpha)<\infty$成立(详细证明见附录).在定理$2$中令$\alpha=\frac{1}{2}$, $f(x, \theta;y, \alpha)=\ln P(x, \theta;y, \alpha)$, 则可得

$ \begin{eqnarray*} E_{\textbf{Q}}[e^{\frac{1}{2}|\ln P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}|X_{i-1}, \xi_{i-1}] &=&\sum\limits_{(y, \alpha)}P(X_{i-1}, \xi_{i-1};y, \alpha)^{-\frac{1}{2}}P(X_{i-1}, \xi_{i-1};y, \alpha)\\ &=&\sum\limits_{(y, \alpha)}P(X_{i-1}, \xi_{i-1};y, \alpha)^{\frac{1}{2}}<\infty . \end{eqnarray*} $

(56)

因此有

$ E_{\textbf{Q}}[e^{\frac{1}{2}|\ln P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}]=E_{\textbf{Q}}[E_{\textbf{Q}}[e^{\frac{1}{2}|\ln P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}|X_{i-1}, \xi_{i-1}]]<\infty. $

(57)

又

$ \begin{eqnarray*} &&E_{\textbf{Q}}[\ln ^{2}P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})e^{\frac{1}{2}|\ln P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})|}|X_{i-1}=x, \xi_{i-1}=\theta]\\ &=&\sum\limits_{(y, \alpha)}\ln ^{2}P(x, \theta;y, \alpha)e^{\frac{1}{2}|\ln P(x, \theta;y, \alpha)|}P(x, \theta;y, \alpha)\\ &<&\sup\limits_{(x, \theta)}\sum\limits_{(y, \alpha)}\ln ^{2}P(x, \theta;y, \alpha)P^{\frac{1}{2}}(x, \theta;y, \alpha)<\infty, \end{eqnarray*} $

(58)

类似于(58)式的讨论, 可得

$ E_{\textbf{Q}}[\ln ^{2}P(X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})e^{\frac{1}{2}|\ln P (X_{i}, \xi_{i};X_{i+1}, \xi_{i+1})|}|X_{i-1}=x, \xi_{i-1}=\theta] <\infty. $

(59)

由(57), (58)和(59)式易知当$f(x, \theta;y, \alpha)=\ln P(x, \theta;y, \alpha)$时满足定理$2$的条件, 故可得

$ \begin{eqnarray*} &\, \, &\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^{n}\ln P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})\\ &=&\sum\limits_{(x, \theta)}\pi(x, \theta)\sum\limits_{(y, \alpha)}\ln P(x, \theta;y, \alpha)P(x, \theta;y, \alpha)\quad \textbf{P}-\text{a.e.}, \ \ \omega\in D(0). \end{eqnarray*} $

(60)

又

$ h(\textbf{P}|\textbf{Q})=0\quad \textbf{P}-\text{a.e.}, \ \ \ \omega\in D(0), $

(61)

则

$ \lim\limits_{n \to \infty}\frac{1}{n}\ln \frac{p(\overrightarrow{X_{0}^{n}}, \overrightarrow{\xi_{0}^{n}})}{q(X_{0})\sum\limits_{i=1}^{n}P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})}=0, \quad \textbf{P}-\text{a.e.}, \ \ \ \omega\in D(0). $

(62)

因此可得

$ \lim\limits_{n \to \infty}\frac{1}{n}\big\{\ln p(\overrightarrow{X_{0}^{n}}, \overrightarrow{\xi_{0}^{n}})-\frac{1}{n}\sum\limits_{i=1}^{n}\ln P(X_{i-1}, \xi_{i-1};X_{i}, \xi_{i})\big\}=0, \quad \textbf{P}-\text{a.e.}, \ \ \ \omega\in D(0). $

(63)

由(2), (60)和(63)知(55)式成立.

命题 令$\sum\limits_{i=1}^{\infty}a_{i}=1,(0<a_{i}<1)$.若$\sum\limits_{i=1}^{\infty}a_{i}^{\frac{1}{2}}\ln^{2}a_{i}<\infty$, 则$\sum\limits_{i=1}^{\infty}a_{i}^{\frac{1}{2}}<\infty$成立.

证 不失一般性, 我们假设$a_{1}>a_{2}>...$, 则存在一个正整数$N$, 使得当$i>N$, 有$a_{i}<\frac{1}{3}$. 易知当$i>N$, 有$\ln^{2}a_{i}>1$. 因此得到 $ \sum\limits_{i>N}a_{i}^{\frac{1}{2}}\ln^{2}a_{i}>\sum\limits_{i>N}a_{i}^{\frac{1}{2}}. $再由正项级数的比较判别法知, $\sum\limits_{i=1}^{\infty}a_{i}^{\frac{1}{2}}<\infty$.