设$ T $是一个无限树, $ \sigma, \tau\; (\sigma\neq \tau) $是$ T $中任意两个顶点, 则存在唯一的从$ \sigma $到$ \tau $的路径, $ \sigma = z_{1}, z_{2}, \cdots, z_{m} = \tau $, 其中$ z_{1}, z_{2}, \cdots, z_{m} $互不相同且$ z_{i}, z_{i+1} $为相邻两顶点, $ m-1 $称为$ \sigma $到$ \tau $的距离.为给$ T $中的顶点编号, 我们选定一个顶点作为根顶点(简称根), 并记为$ o $.对于$ T $中的任意两个顶点$ \sigma, \tau $, 如果$ \tau $是处在从根顶点$ o $到$ \sigma $的唯一路径上, 则记为$ \tau\leq \sigma $.用$ \sigma \wedge \tau $表示同时满足$ \sigma \wedge \tau \leq \tau $与$ \sigma \wedge \tau \leq \sigma $的离$ o $最远的顶点.

本文主要研究二叉树$ T_{C, 2} $ (二叉树$ T_{C, 2} $的根点$ o $与2个支点相连, 其他的支点与3个支点相连(见图 1)).为了方便, 将$ T_{C, 2} $简记为$ T_{2} $.对于$ T_{2} $上的任一顶点$ t $, 记$ |t| $表示根点$ o $和顶点$ t $之间的距离.若$ |t| = n $, 则称$ t $位于树的第$ n $层. $ L_{n} $表示$ T_{2} $的第$ n $层上所有顶点的集合, $ T^{(n)} $表示二叉树$ T_{2} $从$ 0 $层(根)到$ n $层的所有顶点的子图. $ |T^{(n)}| $记为子图$ T^{(n)} $所含顶点数.对于二叉树上任一顶点$ t $, 记$ t^{1} $和$ t^{2} $为$ t $的两个子代.

设$ (\Omega, \mathcal{F}) $为一可测空间, $ \{X_{t}, t\in T_{2}\} $是定义在$ (\Omega, \mathcal{F}) $上且取值于$ G = \{0, 1, \cdots, b-1\} $ ($ b $是正整数)的随机变量集合, 设$ B $为$ T_{2} $的子图, 记$ X^{B} = \{X_{t}, t\in B\} $, $ x^{B} $表示$ X^{B} $的实现.设$ {\bf{P}} $是可测空间$ (\Omega, \mathcal{F}) $上的概率测度, 我们称$ {\bf{P}} $为树$ T_{2} $上的随机场.令$ \{X_{t}, t\in T_{2}\} $在$ {\bf{P}} $下的分布为$ {\bf{P}}(X^{T^{(n)}} = x^{T^{(n)}}) = {\bf{P}}(x^{T^{(n)}}) $.

定义1.1 [二叉树指标分枝马氏链]^[1]  设$ T_{2} $为二叉树, $ \{X_{t}, t\in T_{2}\} $是定义在概率空间$ (\Omega, \mathcal{F}, {\bf{Q}}) $上在$ G = \{0, 1, \cdots, b-1\} $ ($ b $是正整数)中取值的随机变量集合, 设$ p = \{p(x), x\in G\} $是$ G $上一概率分布, $ P = (P(y_{1}, y_{2}|x)) $是定义在$ G\times G^{2} $上的一随机矩阵(满足$ P(y_{1}, y_{2}|x)\geq0, \forall x, y_{1}, y_{2}\in G $及$ \sum\limits_{(y_{1}, y_{2})\in G^{2}}P(y_{1}, y_{2}|x) = 1, \forall x\in G $).如果对$ n\geq1 $, 有

$ \begin{equation} \begin{aligned} {\bf{Q}}(X^{L_{n}} = x^{L_{n}}|X^{T^{(n-1)}} & = x^{T^{(n-1)}}) = \prod\limits_{t\in L_{n-1}}P(x_{t^{1}}, x_{t^{2}}|x_{t}) \end{aligned} \end{equation} $

(1.1)

及

$ {\bf{Q}}(X_{0} = x) = p(x), \forall x\in G, $

则称$ \{X_{t}, t\in T_{2}\} $为具有初始分布$ p $与随机矩阵$ P $并在$ G $中取值的二叉树指标分枝马氏链.

注1.2^[1]  若$ \{X_{t}, t\in T_{2}\} $在概率测度$ {\bf{Q}} $下为二叉树指标齐次分枝马氏链, 则

$ \begin{equation} {\bf{Q}}(X^{T^{(n)}} = x^{T^{(n)}}) = {\bf{Q}}(x^{T^{(n)}}) = p(x_{0})\prod\limits_{t\in T^{(n-1)}}P(x_{t^{1}}, x_{t^{2}}|x_{t}). \end{equation} $

(1.2)

为避免技术问题, 总假定$ p(x) $和$ P(y_{1}, y_{2}|x) $是正的.

定义1.3^[2]  设$ T_{2} $为二叉树, $ \{X_{t}, t\in T_{2}\} $是定义在可测空间$ (\Omega, \mathcal{F}) $上取值于$ G $的随机变量族, $ \forall x, y_{1}, y_{2}\in G $, 设$ p(x)>0 $且$ P = (P(y_{1}, y_{2}|x)) $为正的随机矩阵, 设$ {\bf{Q}}, {\bf{P}} $为$ (\Omega, \mathcal{F}) $上的两个概率测度, 其中$ \{X_{t}, t\in T_{2}\} $在概率测度$ {\bf{Q}} $下为具有初始分布$ p $与随机矩阵$ P $的二叉树指标分枝马氏链.设$ {\bf{P}}(x^{T^{(n)}})>0 $, 令

$ \begin{equation} h({\bf{P}}|{\bf{Q}}) = \limsup\limits_{n\rightarrow \infty}\frac{1}{|T^{(n)}|}\ln\frac{{\bf{P}}(X^{T^{(n)}})}{{\bf{Q}}(X^{T^{(n)}})} = \limsup\limits_{n\rightarrow \infty}\frac{1}{|T^{(n)}|}\ln\frac{{\bf{P}}(X^{T^{(n)}})}{p(X_{0})\prod\limits_{t\in T^{(n-1)}}P(X_{t^{1}}, X_{t^{2}}|X_{t})}, \end{equation} $

(1.3)

则称$ h({\bf{P}}|{\bf{Q}}) $为$ {\bf{P}} $关于$ {\bf{Q}} $的样本相对熵率或渐近对数似然比.

类似于参考文献[2]中引理1的证明可知

$ h({\bf{P}}|{\bf{Q}})\geq0, \; \; \; \; {\bf{P}}-{\rm a.e.}, $

故$ h({\bf{P}}|{\bf{Q}}) $可作为$ T_{2} $上的任意随机场与分枝马氏链之间偏差的一种度量.

树指标随机过程是近年来发展起来的概率论的一个新的研究方向.文献[1]研究了二叉树上分枝马氏链的等价性质; 文献[3]研究了二叉树有限状态分枝马氏链的强大数定律和Shannon-McMillan定理; 文献[4]研究了二叉树上有限状态分枝马氏链的强大数定理; 文献[5]研究了二叉树非齐次分枝马氏链的等价定义、强大数定理和熵遍历定理; 文献[6]研究了关于齐次树上随机场的一类强偏差定理; 文献[2]研究了关于Cayley树上任意随机场和马氏链场的强偏差定理; 文献[7]研究了齐次树指标齐次马氏链随机场的一类强偏差定理; 文献[8]研究了Bethe树指标随机场关于非齐次马氏链的一类强偏差定理.

本文通过引入渐近对数似然比作为二叉树指标任意随机场与分枝马氏链之间偏差的一种度量, 进而构造鞅的方法, 得出了二叉树指标随机场关于分枝马氏链的一类强偏差定理, 作为推论得到了二叉树指标分枝马氏链的强大数定理和渐近均分性.

强偏差定理是由不等式表示的一类强极限定理, 是由等式表示的一类强极限定理的推广.本节将建立二叉树指标任意随机场关于分枝马氏链的一类强偏差定理.

引理2.1  设$ T_{2} $为二叉树, $ \{X_{t}, t\in T_{2}\} $是定义在可测空间$ (\Omega, \mathcal{F}) $上的随机变量族. $ {\bf{P}} $与$ {\bf{Q}} $是$ (\Omega, \mathcal{F}) $上的两个概率测度, 其中$ \{X_{t}, t\in T_{2}\} $在$ {\bf{Q}} $下是具有严格正的随机矩阵$ P = (P(y_{1}, y_{2}|x)) $的二叉树指标分枝马氏链. $ {\bf{P}}(X^{T^{(n)}})>0 $, 设$ \{g_{t}(x, y_{1}, y_{2}), t\in T_{2}\} $是定义在$ G^{3} $上的函数族, $ L_{0} = \{0\} $, $ \mathcal{F}_{n} = \sigma(X^{T^{(n)}}) $, $ n\geq 1 $, 令

$ \begin{equation} \begin{aligned} t_{n}(\lambda, \omega) & = \frac{e^{\lambda \sum\limits_{t\in T^{(n-1)}}g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}} {\prod\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]}\cdot\frac{{\bf{Q}}(X^{T^{(n)}})}{{\bf{P}}(X^{T^{(n)}})}, \end{aligned} \end{equation} $

(2.1)

其中$ \lambda $为实数, $ E_{{\bf{Q}}} $表示在$ {\bf{Q}} $下的数学期望, 则$ \{t_{n}(\lambda, \omega), \mathcal{F}_{n}, n\geq1\} $为在概率测度$ {\bf{P}} $下的非负鞅.

证  由式(1.1), (2.1)可知

$ \begin{equation} \begin{aligned} t_{n}(\lambda, \omega) & = t_{n-1}(\lambda, \omega)\cdot \frac{e^{\lambda \sum\limits_{t\in L_{n-1}}g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}}{\prod\limits_{t\in L_{n-1}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]}\cdot \frac{\prod\limits_{t\in L_{n-1}}P(X_{t^{1}}, X_{t^{2}}|X_{t})}{{\bf{P}}(X^{L_{n}}|X^{T^{(n-1)}})}. \end{aligned} \end{equation} $

(2.2)

令

$ \begin{equation} \begin{aligned} I_{n} & = \frac{e^{\lambda \sum\limits_{t\in L_{n-1}}g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}}{\prod\limits_{t\in L_{n-1}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]}\cdot \frac{\prod\limits_{t\in L_{n-1}}P(X_{t^{1}}, X_{t^{2}}|X_{t})}{{\bf{P}}(X^{L_{n}}|X^{T^{(n-1)}})}, \end{aligned} \end{equation} $

(2.3)

则

$ \begin{eqnarray} &&E_{{\bf{P}}}(I_{n}|\mathcal{F}_{n-1})\\ & = &\sum\limits_{x^{L_{n}}}\frac{e^{\lambda \sum\limits_{t\in L_{n-1}}g_{t}(X_{t}, x_{t^{1}}, x_{t^{2}})}}{\prod\limits_{t\in L_{n-1}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]}\cdot \frac{\prod\limits_{t\in L_{n-1}}P(x_{t^{1}}, x_{t^{2}}|X_{t})}{{\bf{P}}(X^{L_{n}} = x^{L_{n}}|X^{T^{(n-1)}})}\cdot {\bf{P}}(X^{L_{n}} = x^{L_{n}}|X^{T^{(n-1)}})\\ & = &\sum\limits_{x^{L_{n}}}\frac{\prod\limits_{t\in L_{n-1}}e^{\lambda g_{t}(X_{t}, x_{t^{1}}, x_{t^{2}})}P(x_{t^{1}}, x_{t^{2}}|X_{t})}{\prod\limits_{t\in L_{n-1}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]} \end{eqnarray} $

$ \begin{eqnarray} & = &\frac{\prod\limits_{t\in L_{n-1}}\sum\limits_{(x_{t^{1}}, x_{t^{2}})\in G^{2}}e^{\lambda g_{t}(X_{t}, x_{t^{1}}, x_{t^{2}})}\cdot P(x_{t^{1}}, x_{t^{2}}|X_{t})}{\prod\limits_{t\in L_{n-1}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]}\\ & = &\frac{\prod\limits_{t\in L_{n-1}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]}{\prod\limits_{t\in L_{n-1}}E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]} = 1, \; \; \; \; {\bf{P}}-{\rm a.e.}. \end{eqnarray} $

(2.4)

由式(2.2), (2.4)可知

$ E_{{\bf{P}}}(t_{n}(\lambda, \omega)|\mathcal{F}_{n-1}) = t_{n-1}(\lambda, \omega), \; \; \; \; {\bf{P}}-{\rm a.e.}. $

又易知$ E_{{\bf{P}}}[t_{1}(\lambda, \omega)] = 1 $, 因此

$ E_{{\bf{P}}}[t_{n}(\lambda, \omega)] = E_{{\bf{P}}}[E_{{\bf{P}}}(t_{n}(\lambda, \omega)|\mathcal{F}_{n-1})] = E_{{\bf{P}}}[t_{n-1}(\lambda, \omega)] = \cdots = E_{{\bf{P}}}[t_{1}(\lambda, \omega)] = 1, $

故$ \{t_{n}(\lambda, \omega), \mathcal{F}_{n}, n\geq1\} $为在概率测度$ {\bf{P}} $下的非负鞅.

引理2.2  设$ T_{2} $为二叉树, $ \{X_{t}, t\in T_{2}\} $为二叉树指标分枝马氏链, 设$ \{g_{t}(x, y_{1}, y_{2}), t\in T_{2}\} $是定义在$ G^{3} $上的函数族, $ \{a_{n}, n\geq1\} $为正随机变量序列.设$ \alpha>0 $,

$ D(\alpha) = \Big\{\lim\limits_{n}a_{n} = \infty, \limsup\limits_{n}\frac{1}{a_{n}}\sum\limits_{t\in T^{(n-1)}}E[e^{\alpha|g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|}|X_{t}]<\infty\Big\}, $

则

$ \lim\limits_{n}\frac{1}{a_{n}}\sum\limits_{t\in T^{(n-1)}}\{g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})-E[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\} = 0\; \; \; \; {\rm a.e.}, \; \; \; \; \omega\in D(\alpha). $

定理2.3  设$ T_{2} $为二叉树, $ \{X_{t}, t\in T_{2}\} $为定义在可测空间$ (\Omega, \mathcal{F}) $上且取值于$ G = \{0, 1, \cdots, b-1\} $($ b $是正整数)的随机变量族, $ {\bf{P}} $与$ {\bf{Q}} $是定义在$ \mathcal{F} $上的两个概率测度, 并且$ \{X_{t}, t\in T_{2}\} $是在概率测度$ {\bf{Q}} $下的二叉树指标分枝马氏链, 也即式(1.2)成立.令$ h({\bf{P}}|{\bf{Q}}) $如(1.3)式定义, 且$ \{g_{t}(x, y_{1}, y_{2}), t\in T_{2}\} $是定义在$ G^{3} $上的函数族.设$ c\geq0 $是一个常数, 令

$ \begin{equation} \begin{aligned} D(c) & = \{\omega:h({\bf{P}}|{\bf{Q}})\leq c\}. \end{aligned} \end{equation} $

(2.5)

假设存在$ \alpha>0 $, $ \forall i\in G $, 有

$ b_{\alpha}(i) = \limsup\limits_{n\rightarrow \infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[e^{\alpha|g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|}|X_{t} = i]\leq \tau. $

令

$ \begin{equation} \begin{aligned} H_{\beta}^{\alpha} & = \frac{2\tau}{e^{2}(\beta-\alpha)^{2}}, \\ \end{aligned} \end{equation} $

(2.6)

其中$ 0<\beta<\alpha $.则当$ 0\leq c\leq \beta^{2}H_{\beta}^{\alpha} $时, 有

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow \infty}\frac{1}{|T^{(n)}|}\left|\sum\limits_{t\in T^{(n-1)}}\{g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})-E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\right|\\ \leq&2\sqrt{cH_{\beta}^{\alpha}}, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(c). \end{aligned} \end{equation} $

(2.7)

特别地, 有

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow \infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})-E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\} = 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(0). \end{aligned} \end{equation} $

(2.8)

证  令$ t_{n}(\lambda, \omega) $如(2.1)式定义.由引理2.1知, $ \{t_{n}(\lambda, \omega), \mathcal{F}_{n}, n\geq1\} $是非负$ {\bf{P}} $ -鞅.则由Doob鞅收敛定理可知

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}t_{n}(\lambda, \omega) & = t(\lambda, \omega)<\infty, \; \; \; \; {\bf{P}}-{\rm a.e.}.\\ \end{aligned} \end{equation} $

(2.9)

因此有

$ \limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\ln t_{n}(\lambda, \omega)\leq0, \; \; \; \; {\bf{P}}-{\rm a.e.}. $

由式(2.1), (2.9)可知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\ln t_{n}(\lambda, \omega)\\ = &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\{\sum\limits_{t\in T^{(n-1)}}[\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})-\ln E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]+\ln\frac{{\bf{Q}}(x^{T^{(n)}})}{{\bf{P}}(x^{T^{(n)}})}]\}\\ \leq&0, \; \; \; \; {\bf{P}}-{\rm a.e.}.\\ \end{aligned} \end{equation} $

(2.10)

由式(1.3)及(2.10)可知

$ \begin{equation} \begin{aligned} \limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})-\ln E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]\} \leq h({\bf{P}}|{\bf{Q}}), \; \; \; {\bf{P}}-{\rm a.e.}. \end{aligned} \end{equation} $

(2.11)

由式(2.11)可知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{\lambda}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ = &\limsup\limits_{n\rightarrow\infty}\frac{\lambda}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})-\frac{1}{\lambda}\ln E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]\}\\ &+\limsup\limits_{n\rightarrow\infty}\frac{\lambda}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{\frac{1}{\lambda}\ln E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]-E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ \leq& \limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{\ln E_{{\bf{Q}}}[e^{\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})}|X_{t}]-E_{{\bf{Q}}}[\lambda g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ &+ h({\bf{P}}|{\bf{Q}}), \; \; \; \; {\bf{P}}-{\rm a.e.}. \end{aligned} \end{equation} $

(2.12)

令$ |\lambda|\leq \beta $.由不等式$ \ln x \leq x-1\; \; (x> 0) $和$ e^{x}-1-x\leq\frac{x^{2}}{2}e^{|x|} $, 并注意到

$ \max\{x^{2}e^{-hx}, x\geq0\} = \frac{4}{e^{2}h^{2}}\; \; \; \; (h>0), $

则对于所有的$ \omega\in \Omega $, 有

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{\ln E_{{\bf{Q}}}[e^{\lambda g_{t}}|X_{t}]-E_{{\bf{Q}}}[\lambda g_{t}|X_{t}]\}\\ \leq& \limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ E_{{\bf{Q}}}[e^{\lambda g_{t}}|X_{t}]-1-E_{{\bf{Q}}}[\lambda g_{t}|X_{t}]\}\\ = &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[e^{\lambda g_{t}}-1-\lambda g_{t}|X_{t}]\\ \leq& \limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[\frac{1}{2}\lambda^{2}g_{t}^{2}e^{|\lambda g_{t}|}|X_{t}]\\ = &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[\frac{1}{2}\lambda^{2}g_{t}^{2}e^{\alpha|g_{t}|}e^{(|\lambda|-\alpha)|g_{t}|}|X_{t}]\\ \leq& \frac{\lambda^{2}}{2}\limsup\limits_{n\rightarrow\infty}\sum\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[\frac{4}{e^{2}(|\lambda|-\alpha)^{2}}e^{\alpha|g_{t}|}|X_{t}]\\ \leq& \frac{2\lambda^{2}\tau}{e^{2}(\beta-\alpha)^{2}}. \end{aligned} \end{equation} $

(2.13)

其中为了方便, 将用$ g_{t} $代替$ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}}) $, 由式(2.6), (2.12)和(2.13)知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{\lambda}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ \leq& \lambda^{2}H_{\beta}^{\alpha}+h({\bf{P}}|{\bf{Q}}), \; \; \; \; {\bf{P}}-{\rm a.e.}. \end{aligned} \end{equation} $

(2.14)

由式(2.5)与(2.14)可知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{\lambda}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ \leq& \lambda^{2}H_{\beta}^{\alpha}+c, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(c). \end{aligned} \end{equation} $

(2.15)

当$ 0<\lambda\leq\beta<\alpha $时, 将式(2.15)左右两边同时除以$ \lambda $可知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ \leq& \lambda H_{\beta}^{\alpha}+\frac{c}{\lambda}, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(c). \end{aligned} \end{equation} $

(2.16)

容易看出, 当$ 0<c<\beta^{2}H_{\beta}^{\alpha} $时, 函数$ f(\lambda) = \lambda H_{\beta}^{\alpha}+\frac{c}{\lambda} $在$ \lambda = \sqrt{\frac{c}{H_{\beta}^{\alpha}}} $时获得最小值$ f(\sqrt{\frac{c}{H_{\beta}^{\alpha}}}) = 2\sqrt{cH_{\beta}^{\alpha}} $.令式(2.16)中$ \lambda = \sqrt{\frac{c}{H_{\beta}^{\alpha}}} $, 可知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ \leq& 2\sqrt{cH_{\beta}^{\alpha}}, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(c). \end{aligned} \end{equation} $

(2.17)

当$ c = 0 $, 由式(2.16)可知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ &\leq \lambda H_{\beta}^{\alpha}, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(0). \end{aligned} \end{equation} $

(2.18)

令式(2.18)中$ \lambda\rightarrow 0^{+} $, 可知

$ \begin{equation} \begin{aligned} &\limsup\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ \leq& 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(0). \end{aligned} \end{equation} $

(2.19)

因此当$ c = 0 $时, 公式(2.17)成立.当$ -\alpha<-\beta\leq\lambda<0 $时, 由式(2.15)类似可知

$ \begin{equation} \begin{aligned} &\liminf\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\{ g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})- E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}]\}\\ \geq& -2\sqrt{cH_{\beta}^{\alpha}}, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega\in D(c). \end{aligned} \end{equation} $

(2.20)

由式(2.17), (2.20)可知式(2.7), 继而可知式(2.8).故此定理2.3得证.

本节研究二叉树指标随机场关于分枝马氏链的强大数定理与渐近均分性(AEP), 作为推论得到了二叉树指标分枝马氏链的强大数定理和渐近均分性.

设$ T_{2} $为二叉树, $ \{X_{t}, t\in T_{2}\} $是定义在可测空间$ (\Omega, \mathcal{F}) $上在$ G $中取值的随机变量族, 可得结论如下.

定义3.1^[3]  设$ {\bf{P}} $是$ (\Omega, \mathcal{F}) $上的一个概率测度, $ x^{T^{(n)}} $为$ X^{T^{(n)}} $的实现.记$ \{X_{t}, t\in T_{2}\} $在概率测度$ {\bf{P}} $下的分布为$ {\bf{P}}(X^{T^{(n)}} = x^{T^{(n)}}) = {\bf{P}}(x^{T^{(n)}})>0 $.设

$ \begin{equation} \begin{aligned} f_{n}(\omega) & = -\frac{1}{|T^{(n)}|}\ln {\bf{P}}(X^{T^{(n)}}), \end{aligned} \end{equation} $

(3.1)

则称$ f_{n}(\omega) $为$ X^{T^{(n)}} $在概率测度$ {\bf{P}} $下的熵密度.如果$ \{X_{t}, t\in T_{2}\} $在概率测度$ {\bf{Q}} $下是具有初始分布$ p(x) $和随机矩阵$ P $的二叉树指标分枝马氏链, 则

$ \begin{equation} \begin{aligned} g_{n}(\omega) & = -\frac{1}{|T^{(n)}|}\ln {\bf{Q}}(X^{T^{(n)}}) = -\frac{1}{|T^{(n)}|}[\ln p(X_{0})+\sum\limits_{t\in T^{(n-1)}}\ln P(X_{t^{1}}, X_{t^{2}}|X_{t})].\\ \end{aligned} \end{equation} $

(3.2)

熵密度$ f_{n}(\omega) $收敛($ L_{1} $收敛, 依概率收敛, a.e.收敛)到一个常数的性质在信息论中被称为渐近均分性(AEP)或者是Shannon-McMillan定理.

定义3.2^[3]  设$ S_{k}(T^{(n)})(k\in G) $是随机变量集$ \{X_{t}, t\in T^{(n)}\} $中$ k $出现的次数, $ S_{n}(k, l_{1}, l_{2}) = S_{k, l_{1}, l_{2}}(T^{(n)}) $是随机序偶集$ \{(X_{t}, X_{t^{1}}, X_{t^{2}}), t\in T^{(n-1)}\} $中序偶$ (k, l_{1}, l_{2}) $ $ (k, l_{1}, l_{2}\in G) $出现的次数, 即

$ S_{k}^{1}(T^{(n)}) = |\{t\in T^{(n-1)}, X_{t^{1}} = k\}| = \sum\limits_{t\in T^{(n-1)}}I_{k}(X_{t^{1}}), $

$ \begin{eqnarray*} &&S_{k}^{2}(T^{(n)}) = |\{t\in T^{(n-1)}, X_{t^{2}} = k\}| = \sum\limits_{t\in T^{(n-1)}}I_{k}(X_{t^{2}}), \\ &&S_{k}(T^{(n)}) = |\{t\in T^{(n)}, X_{t} = k\}| = \sum\limits_{t\in T^{(n)}}I_{k}(X_{t}), \\ &&S_{n}(k, l_{1}, l_{2}) = |\{t\in T^{(n-1)}, (X_{t}, X_{t^{1}}, X_{t^{2}}) = (k, l_{1}, l_{2})\}|. \end{eqnarray*} $

显然

$ \begin{equation} \begin{aligned} &S_{k}(T^{(n)}) = S_{k}^{1}(T^{(n)})+S_{k}^{2}(T^{(n)})+I_{k}(X_{0}), \\ &S_{n}(k, l_{1}, l_{2}) = \sum\limits_{t\in T^{(n-1)}}I_{k}(X_{t})I_{l_{1}}(X_{t^{1}})I_{l_{2}}(X_{t^{2}}), \nonumber \end{aligned} \end{equation} $

其中$ I_{k}(i) = \begin{cases} 1, & i = k, \\ 0, & i\neq k. \end{cases} $

定理3.3  设$ {\bf{P}} $与$ {\bf{Q}} $是定义在$ \mathcal{F} $上的两个概率测度, 并且$ \{X_{t}, t\in T_{2}\} $是由定义1.1定义的二叉树指标分枝马氏链$ S_{k}^{1}(T^{(n)}) $, $ S_{k}^{2}(T^{(n)}) $, $ S_{k}(T^{(n)}) $如定义3.2所定义, 令

$ P_{1}(y_{1}|x) = \sum\limits_{y_{2}}P(y_{1}, y_{2}|x), \; \; \; \; P_{2}(y_{2}|x) = \sum\limits_{y_{1}}P(y_{1}, y_{2}|x). $

设转移矩阵$ P_{1} = (P_{1}(y_{1}|x)), P_{2} = (P_{2}(y_{2}|x)) $, 令转移矩阵$ R = \frac{1}{2}(P_{1}+P_{2}) $, 则

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}\frac{S_{k}(T^{(n)})}{|T^{(n)}|} & = \pi(k), \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0), \end{aligned} \end{equation} $

(3.3)

其中$ (\pi(0), \cdots, \pi(b-1)) $为由矩阵$ R $决定的平稳分布.

证  首先转移矩阵$ R = \frac{1}{2}(P_{1}+P_{2}) $是遍历的, 这是因为$ P(y_{1}, y_{2}|x)>0 $, 可知$ P_{1}(y_{1}|x)>0, $ $ P_{2}(y_{2}|x)>0, $所以$ \frac{1}{2}(P_{1}(y_{1}|x)+P_{2}(y_{2}|x))>0 $.即$ R $是遍历的.

令$ g_{t}(x, y_{1}, y_{2}) = I_{k}(y_{1}) $, 则

$ \begin{equation*} \begin{aligned} &\sum\limits_{t\in T^{(n-1)}}g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}}) = \sum\limits_{t\in T^{(n-1)}}I_{k}(X_{t^{1}}) = S_{k}^{1}(T^{(n)}), \\ &\sum\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}] = \sum\limits_{t\in T^{(n-1)}}\sum\limits_{(x_{t^{1}}, x_{t^{2}})\in G^{2}}g_{t}(X_{t}, x_{t^{1}}, x_{t^{2}})P(x_{t^{1}}, x_{t^{2}}|X_{t})\\ = &\sum\limits_{t\in T^{(n-1)}}\sum\limits_{(x_{t^{1}}, x_{t^{2}})\in G^{2}}I_{k}(x_{t^{1}})P(x_{t^{1}}, x_{t^{2}}|X_{t}) = \sum\limits_{t\in T^{(n-1)}}P_{1}(k|X_{t}) = \sum\limits_{t\in T^{(n-1)}}\sum\limits_{l = 0}^{b-1}I_{l}(X_{t})P_{1}(k|l)\\ = &\sum\limits_{l = 0}^{b-1}S_{l}(T^{(n-1)})P_{1}(k|l), \end{aligned} \end{equation*} $

故由式(2.8)可知

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}[S_{k}^{1}(T^{(n)})-\sum\limits_{l = 0}^{b-1}S_{l}(T^{(n-1)})P_{1}(k|l)] & = 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0). \end{aligned} \end{equation} $

(3.4)

同理令$ g_{t}(x, y_{1}, y_{2}) = I_{k}(y_{2}) $.类似可知

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}[S_{k}^{2}(T^{(n)})-\sum\limits_{l = 0}^{b-1}S_{l}(T^{(n-1)})P_{2}(k|l)] & = 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0). \end{aligned} \end{equation} $

(3.5)

由式(3.5)与式(3.6)相加, 结合$ \lim\limits_{n\rightarrow\infty}\frac{|T^{(n)}|}{|T^{(n-1)}|} = 2, \; \; R = \frac{1}{2}(P_{1}+P_{2}), $可知

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}\{\frac{S_{k}(T^{(n)})}{|T^{(n)}|}-\sum\limits_{l = 0}^{b-1}\frac{S_{l}(T^{(n-1)})}{|T^{(n-1)}|}R(k|l)\} & = 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0). \end{aligned} \end{equation} $

(3.6)

用$ R(m|k) $乘以式(3.7), 将对$ k\in \{0, 1，\cdots, b-1\} $求和, 并再次利用式(3.7)可知

$ \begin{equation} \begin{aligned} &\lim\limits_{n\rightarrow\infty}\{\sum\limits_{k = 0}^{b-1}\frac{S_{k}(T^{(n)})R(m|k)}{|T^{(n)}|}-\frac{S_{m}(T^{(n+1)})}{|T^{(n+1)}|}+\frac{S_{m}(T^{(n+1)})}{|T^{(n+1)}|}-\sum\limits_{l = 0}^{b-1}\frac{S_{l}(T^{(n-1)})}{|T^{(n-1)}|}R^{(2)}(m|l)\}\\ = &\lim\limits_{n\rightarrow\infty}\{\{\frac{S_{m}(T^{(n+1)})}{|T^{(n+1)}|}-\sum\limits_{l = 0}^{b-1}\frac{S_{l}(T^{(n-1)})}{|T^{(n-1)}|}R^{(2)}(m|l)\} = 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0). \end{aligned} \end{equation} $

(3.7)

由归纳法可知

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}\{\frac{S_{m}(T^{(n+N)})}{|T^{(n+N)}|}-\sum\limits_{l = 0}^{b-1}\frac{S_{l}(T^{(n-1)})}{|T^{(n-1)}|}R^{(N+1)}(m|l)\} & = 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0). \end{aligned} \end{equation} $

(3.8)

注意到

$ \sum\limits_{l = 0}^{b-1}\frac{S_{l}(T^{(n-1)})}{|T^{(n-1)}|} = \frac{1}{|T^{(n-1)}|}\sum\limits_{l = 0}^{b-1}S_{l}(T^{(n-1)}) = 1, $

且$ \lim\limits_{N\rightarrow\infty}R^{(N)}(k|l) = \pi(k), \; \; k\in G, $则式(3.4)得证.

定理3.4  在定理3.3的条件下, 令$ S_{n}(k, l_{1}, l_{2}) = S_{k, l_{1}, l_{2}}(T^{(n)}) $如定义3.2所述, 则

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}\frac{S_{n}(k, l_{1}, l_{2})}{|T^{(n)}|} & = \frac{1}{2}\pi(k)P(l_{1}, l_{2}|k), \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0), \end{aligned} \end{equation} $

(3.9)

其中$ (\pi(0), \cdots, \pi(b-1)) $为由矩阵$ R $决定的平稳分布.

证  令$ g_{t}(x, y_{1}, y_{2}) = I_{k}(x)I_{l_{1}}(y_{1})I_{l_{2}}(y_{2}) $, 则

$ \begin{equation} \begin{aligned} \sum\limits_{t\in T^{(n-1)}}g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})& = \sum\limits_{t\in T^{(n-1)}}I_{k}(X_{t})I_{l_{1}}(X_{t^{1}})I_{l_{2}}(X_{t^{2}}) = S_{n}(k, l_{1}, l_{2}), \\ \sum\limits_{t\in T^{(n-1)}}E_{{\bf{Q}}}[g_{t}(X_{t}, X_{t^{1}}, X_{t^{2}})|X_{t}] & = \sum\limits_{t\in T^{(n-1)}}\sum\limits_{(x_{t^{1}}, x_{t^{2}})\in G^{2}}g_{t}(X_{t}, x_{t^{1}}, x_{t^{2}})P(x_{t^{1}}, x_{t^{2}}|X_{t})\\ & = \sum\limits_{t\in T^{(n-1)}}\sum\limits_{(x_{t^{1}}, x_{t^{2}})\in G^{2}}I_{k}(X_{t})I_{l_{1}}(x_{t^{1}})I_{l_{2}}(x_{t^{2}})P(x_{t^{1}}, x_{t^{2}}|X_{t})\\ & = \sum\limits_{t\in T^{(n-1)}}I_{k}(X_{t})\sum\limits_{(x_{t^{1}}, x_{t^{2}})\in G^{2}}I_{l_{1}}(x_{t^{1}})I_{l_{2}}(x_{t^{2}})P(x_{t^{1}}, x_{t^{2}}|X_{t})\\ & = \sum\limits_{t\in T^{(n-1)}}I_{k}(X_{t})P(l_{1}, l_{2}|k) = S_{n-1}(k)P(l_{1}, l_{2}|k). \end{aligned} \end{equation} $

(3.10)

由式(2.8)可知

$ \lim\limits_{n\rightarrow\infty}\frac{1}{|T^{(n)}|}[S_{n}(k, l_{1}, l_{2})-S_{n-1}(k)P(l_{1}, l_{2}|k)] = 0, \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0). $

因为$ \lim\limits_{n\rightarrow\infty}\frac{|T^{(n-1)}|}{|T^{(n)}|} = \frac{1}{2}, $所以

$ \lim\limits_{n\rightarrow\infty}\frac{S_{n-1}(k)}{|T^{(n)}|} = \lim\limits_{n\rightarrow\infty}\frac{S_{n-1}(k)}{|T^{(n-1)}|}\frac{|T^{(n-1)}|}{|T^{(n)}|} = \frac{1}{2}\pi(k), $

因此式(3.10)得证.

定理3.5  在定理3.3的条件下, $ f_{n}(\omega) $是如(3.1)所定义的熵密度, 则

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}f_{n}(\omega) = -\sum\limits_{k\in G}\sum\limits_{l_{1}\in G}\sum\limits_{l_{2}\in G}\frac{1}{2}\pi(k)P(l_{1}, l_{2}|k)\ln P(l_{1}, l_{2}|k), \; \; \; \; {\bf{P}}-{\rm a.e.}, \; \; \; \; \omega \in D(0), \end{aligned} \end{equation} $

(3.11)

其中$ (\pi(0), \cdots, \pi(b-1)) $为由矩阵$ R $决定的平稳分布.

证  因为

$ \begin{equation} \begin{aligned} &-\frac{1}{|T^{(n)}|}\sum\limits_{t\in T^{(n-1)}}\ln P(X_{t^{1}}, X_{t^{2}}|X_{t}) & = -\lim\limits_{n\rightarrow\infty}\sum\limits_{k\in G}\sum\limits_{l_{1}\in G}\sum\limits_{l_{2}\in G}\ln P(l_{1}, l_{2}|k)\frac{S_{n}(k, l_{1}, l_{2})}{|T^{(n)}|}, \end{aligned} \end{equation} $

(3.12)

由式(3.2)与式(3.10)、(3.13)可知

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}g_{n}(\omega) & = -\sum\limits_{k\in G}\sum\limits_{l_{1}\in G}\sum\limits_{l_{2}\in G}\frac{1}{2}\pi(k)P(l_{1}, l_{2}|k)\ln P(l_{1}, l_{2}|k), \; \; \; \; {\bf{P}}-{\rm a.e.}\; \; \; \; \omega \in D(0). \end{aligned} \end{equation} $

(3.13)

由式(1.3), (3.1), (3.2)可知

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}g_{n}(\omega) & = \lim\limits_{n\rightarrow\infty}f_{n}(\omega), \; \; \; \; {\bf{P}}-{\rm a.e.}\; \; \; \; \omega \in D(0). \end{aligned} \end{equation} $

(3.14)

故综合式(3.14), (3.15)可知式(3.12)得证.

定理3.6  设$ \{X_{t}, t\in T_{2}\} $在概率测度$ {\bf{Q}} $下是具有严格正的随机矩阵$ P = (P(y_{1}, y_{2}|x)) $的二叉树指标分枝马氏链, $ \{X_{t}, t\in T_{2}\}, S_{n}(k), S_{n}(k, l_{1}, l_{2}), $ $ f_{n}(\omega) $如定义3.2所述, 令

$ P_{1}(y_{1}|x) = \sum\limits_{y_{2}}P(y_{1}, y_{2}|x), P_{2}(y_{2}|x) = \sum\limits_{y_{1}}P(y_{1}, y_{2}|x). $

设转移矩阵$ P_{1} = (P_{1}(y_{1}|x))，P_{2} = (P_{2}(y_{2}|x)) $, 假设转移矩阵$ R = \frac{1}{2}(P_{1}+P_{2}) $, 则

$ \begin{equation} \begin{aligned} \lim\limits_{n\rightarrow\infty}\frac{S_{n}(k)}{|T^{(n)}|} & = \pi(k), \; \; \; \; {\bf{Q}}-{\rm a.e.}, \\ \lim\limits_{n\rightarrow\infty}\frac{S_{n}(k, l_{1}, l_{2})}{|T^{(n)}|} & = \frac{1}{2}\pi(k)P(l_{1}, l_{2}|k), \; \; \; \; {\bf{Q}}-{\rm a.e.}, \\ \lim\limits_{n\rightarrow\infty}f_{n}(\omega) & = -\sum\limits_{k\in G}\sum\limits_{l_{1}\in G}\sum\limits_{l_{2}\in G}\frac{1}{2}\pi(k)P(l_{1}, l_{2}|k)\ln P(l_{1}, l_{2}|k), \; \; \; \; {\bf{Q}}-{\rm a.e.}, \end{aligned} \end{equation} $

(3.15)

其中$ (\pi(0), \cdots, \pi(b-1)) $为由矩阵$ R $决定的平稳分布.

证  令定理3.3中$ {\bf{P}}\equiv {\bf{Q}} $, 则$ g_{n}(\omega) = f_{n}(\omega) $, 进而知$ D(0) = \Omega $.因此由式(3.4), (3.10), (3.12)可知式(3.16), (3.17), (3.18)得证.