张志洋, 胡晓予
中国科学院大学数学科学学院, 北京 100049
2015年07月08日 收稿; 2016年03月24日 收修改稿
基金项目: 国家自然科学基金（11171342）资助
通信作者: 胡晓予?E-mail:xyhu@ucas.ac.cn

摘要: 研究两种情形下二重随机变量序列$\left\{ {{X_{n,i}}:n \ge 1,i \ge 1} \right\}$的随机和的P-收敛性.一种情形是求和个数（随机变量ξ_n）与$\left\{ {{X_{n,i}}:n \ge 1,i \ge 1} \right\}$相互独立，另一种情形是$\left\{ {{\xi _n}:n \ge 1} \right\}$与$\left\{ {{X_{n,i}}:n \ge 1,i \ge 1} \right\}$不一定独立.这其中包括变化环境中分枝过程的P-收敛性.
关键词: 二重序列随机和分枝过程P-收敛
P-convergence of a double sequence of random variables
ZHANG Zhiyang, HU Xiaoyu
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China


Abstract: In this study we investigate the P-convergence of random sum(say ξ_n) of a double random sequence of random variables $\left\{ {{X_n}:n \ge 1,i \ge 1} \right\}$ in two cases. The first case is that $\left\{ {{\xi _n}:n \ge 1} \right\}$ is independent of $\left\{ {{X_{n,i}}:n \ge 1,i \ge 1} \right\}$, and the second case is that $\left\{ {{\xi _n}:n \ge 1} \right\}$ is not necessarily independent of the double sequence $\left\{ {{X_{n,i}}:n \ge 1,i \ge 1} \right\}$.These results include P-convergence of a branching process in varying environments.
Key words: random sum of a double random sequence of random variablesbranching processP-convergence
极限理论是概率论中备受关注的问题之一, 中心极限定理和大数律的研究, 可上溯至十八世纪初, 不过那时的结果主要针对的是单序列的部分和(求和的个数是确定的). Kolmogorov在20世纪三四十年代, 对行内独立的二重序列确定和的极限定理(包括中心极限定理和重对数律等)做了系统的研究, 并得出一系列奠基性的结果^[1-2].
在此之后, 由于实际的需要, 中心极限定理的研究对象开始从单纯的行内独立的随机变量序列的确定和转向随机个随机变量之和.如胡迪鹤^[3]通过单序列随机和的中心极限定理结果, 将Donsker不变原理推广到分枝过程的情形.
1970—1971年, Heyde^[4]得到关于分枝过程的一系列极限定理结果, 其中包括中心极限定理和重对数律.其后一些工作讨论了随机环境中分枝过程的极限理论^[5-6].
1 背景知识本文讨论一致渐进可忽略(u.a.n.)二重随机变量序列的随机和的P-收敛性, 包括变化环境中的分枝过程的P-收敛性.对二重随机变量序列的确定和而言, P-收敛性的讨论通常是弱收敛研究的铺垫^[2].
令$(\mathit{\Omega} ,\mathscr{F},P)$为一个概率空间, 本文的所有随机变量均定义在此空间上.文中总记$\mathbb{N}$为全体非负整数构成的集合, ${\mathbb{N}^ + }$为全体正整数构成的集合.
定义1.1 ??令Z₀=1且对于任意的n≥0,

${Z_{n + 1}} = \left\{ \begin{array}{l}\sum\limits_{j = 1}^{{Z_n}} {{X_{n,j}}\left( x \right),\;{Z_n} \ne 0;} \\0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Z_n} = 0,\end{array} \right.$

其中，$\left\{ {{X_{0,1}},{X_{n,j}}:n \ge 0,j \ge 1} \right\}$相互独立. X_{0, 1}的分布为$\left\{ {{p_0}\left( j \right),j \in \mathbb{N}} \right\}$, 对于任意的$n \ge 1,\left\{ {{X_{n,j}}:j \in \mathbb{N}} \right\}$有相同的分布$\left\{ {{p_n}\left( k \right),k \in \mathbb{N}} \right\},\left\{ {{Z_n}:n \ge 0} \right\}$称为变化环境中的分枝过程.记μ₀=E(X_{0, 1}), μ_n=E(X_{n, j}), n≥1, 又记m₀=μ₀, m_n=E(Z_n)=μ₀…μ_n-1, 当1 < μ_n < ∞时, 称该分枝过程为上临界的.在本文中总假定p_n(0)=0, 0 < p_n(1) < 1.特别地, 当p_n(k)=p(k), n≥0, 则{Z_n:n≥0}为经典的G-W分枝过程.
定义1.2 ??设$\left\{ {{\xi _n}:n \ge 1} \right\}$为一列定义在$\left( {\mathit{\Omega} ,\mathscr{F},P} \right)$上且取值于$\mathbb{N}$的随机变量序列, 又设ξ_n→∞a.s.当n→∞, 称{V_{n, k}:n≥1, k≥1}为关于{ξ_n:n≥1}的一致渐进可忽略序列, 简称u.a.n.序列, 如果它满足下面的条件

$\forall \epsilon > 0,\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} P\left( {\left| {{V_{n,k}}} \right| \ge \epsilon} \right) = 0\;\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$

注1.1 ??当ξ_n≡k_n时(k_n为正整数, $\forall n \in {\mathbb{N}^ + }$), 则{V_{n, k}:n≥1, k≥1}就是经典意义下的u.a.n.序列(见文献[7]).
引理1.1 ??沿用前面的符号, 设{Z_n, n≥0}为定义1.1中所定义的变化环境中的上临界的分枝过程, {X_{n, j}:j≥1}的期望为μ_n.假定Z_n→∞a.s.当$n \to \infty ,E\left| {{X_{n,j}} - {\mu _n}} \right| \le M < \infty ,\;\forall j,n \in {\mathbb{N}^ + }$, 则对任意单调上升函数f且limx→∞f(x)=+∞, 如下序列

${Y_{n,j}}: = \frac{{{X_{n,j}} - {\mu _n}}}{{f\left( {{Z_n}} \right)}},$

是关于{Z_n, n≥0}的u.a.n.序列.
证明 ??首先根据上临界分枝过程的性质和p_n(0)≡0的假设知: Z_n→∞a.s., 当n→∞.
现在只需要对于任意的$ \epsilon > 0$验证下式

$\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le j \le {Z_n}} P\left( {\frac{{\left| {{X_{n,j}} - {\mu _n}} \right|}}{{f\left( {{Z_n}} \right)}} \ge \epsilon} \right) = 0\;{\rm{a}}{\rm{.s}}{\rm{.}}$

给定$1 > \epsilon > 0$, 由于$\mathop {\lim }\limits_{n \to \infty } {Z_n} = \infty \;{\rm{a}}{\rm{.s}}{\rm{.}}$, 故对于任意的η>0, C>0, 总存在Ω₀Ω以及n₀>0使得P(Ω₀)>1-η且f(Z_n)>C, ω∈Ω₀, n≥n₀(此处Ω₀和n₀仅依赖于η和C).由Chebyshev不等式有:当n≥n₀时,

$\begin{array}{l}P\left( {\frac{{\left| {{X_{n,j}} - {\mu _n}} \right|}}{{f\left( {{Z_n}} \right)}} \ge \epsilon} \right)\\ \le P\left( {\frac{{\left| {{X_{n,j}} - {\mu _n}} \right|}}{{f\left( {{Z_n}} \right)}} \ge \epsilon,{\mathit{\Omega} _0}} \right) + \eta \\ \le \frac{1}{{\epsilon f\left( C \right)}}E\left( {\left| {{X_{n,j}} - {\mu _n}} \right|} \right) + \eta \\ \le \frac{M}{{\epsilon f\left( C \right)}} + \eta ,\end{array}$

令C→∞, 则有对a.s.ω

$\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le j \le {Z_n}} P\left( {\frac{{\left| {{X_{n,j}}{\mu _n}} \right|}}{{f\left( {{Z_n}} \right)}} \ge \epsilon} \right) \le \eta ,$

而η是任意的, 故$\left\{ {{Y_{n,j}}:n \ge 1,j \ge 1} \right\}$是关于$\left\{ {{Z_n},n \ge 0} \right\}$的u.a.n.序列.
注1.2 ??类似于上述证明可得, 在同样条件下

$\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le j \le {Z_n}} P\left( {\frac{{\left| {{X_{n,j}}{\mu _n}} \right|}}{{f\left( {{Z_n}} \right)}} \ge \epsilon\left| {{Z_n}} \right.} \right) = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$

此时称$\left\{ {\frac{{\left| {{X_{n,j}} - {\mu _n}} \right|}}{{f\left( {{Z_n}} \right)}}:n \ge 1,j \ge 1} \right\}$为关于{Z_n, n≥0}的条件u.a.n.序列.
注1.3 ??令Z_{n, k}^(j)为定义1.1中的分枝过程第n代第j个个体的第k代子孙的数量, 记$W_n^{\left( j \right)}: = \mathop {\min }\limits_{n \to \infty } Z_{n,k}^{\left( j \right)}{\left( {\prod\limits_{i = n + 1}^{n + k} {{\mu _i}} } \right)^{ - 1}}$(详见Athreya和Ney^[8]和高振龙^[5]), 仿照上述证明可知, 当$E\left| {1 - W_n^{\left( j \right)}} \right| \le K < \infty $时, ${U_{n,j}}: = \frac{{1 - W_n^{\left( j \right)}}}{{\sqrt {{Z_n}} }}$也为关于$\left\{ {{Z_n},n \ge 0} \right\}$的u.a.n.序列. Heyde^[4]讨论了$\left\{ {{U_{n,j}},n \ge 1,j \ge 1} \right\}$的中心极限定理.
2 主要结果命题2.1 ??设$\left\{ {{V_{n,k}}:n \ge 1,k \ge 1} \right\}$为关于$\left\{ {{\xi _n}:n \ge 1} \right\}$的u.a.n.序列, 记F_{n, k}(t)为V_{n, k}的分布函数, f_{n, k}(t)为V_{n, k}的特征函数, 则下面6个条件等价:
1)$\forall \epsilon > 0,\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} P\left( {\left| {{V_{n,k}}} \right| \ge \epsilon} \right) = 0\;\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$;
2)在任意的有限区间[a, b]上对a.s.ω有, $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} {\mathop{\rm Re}\nolimits} \left( {1 - {f_{n,k}}\left( t \right)} \right)0\forall t \in \left[ {a,b} \right]$, 其中Re(1-f_{n, k}(t))为1-f_{n, k}(t)的实部, 上述收敛对于t∈[a, b]是一致的;
3)在任意的有限区间[a, b]上对a.s.ω有, $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} \left| {1 - {f_{n,k}}\left( t \right)} \right| = 0\forall t \in \left[ {a,b} \right]$, 上述收敛对于t∈[a, b]是一致的;
4)对任意的t, $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} \left| {1 - {f_{n,k}}\left( t \right)} \right| = 0\;{\rm{a}}{\rm{.s}}{\rm{.}}$;
5)对任意的t, $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} {\mathop{\rm Re}\nolimits} \left( {1 - {f_{n,k}}\left( t \right)} \right) = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$;
6)对任意的t, $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} \int_{ - \infty }^\infty {\frac{{{x^2}}}{{1 + {x^2}}}{\rm{d}}{F_{n,k}}\left( x \right) = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}} $.
证明 ??1)$ \Rightarrow $2).选取T>0使得$\left[ {a,b} \right] \subset \left[ { - T,T} \right]$, 给定$ \epsilon > 0$, 对于任意的$t \in \left[ { - T,T} \right]$，

$\begin{array}{c}{\mathop{\rm Re}\nolimits} \left( {1 - {f_{n,k}}\left( t \right)} \right)\\ = \int_{ - \infty }^\infty {\left( {1 - \cos xt} \right){\rm{d}}{F_{n,k}}\left( x \right)} \\ = \int_{\left| x \right| < \epsilon } {\left( {1 - \cos \;xt} \right){\rm{d}}{F_{n,k}}\left( x \right) + } \\\int_{\left| x \right| \ge \epsilon} {\left( {1 - \cos xt} \right){\rm{d}}{F_{n,k}}\left( x \right)} \\ \le \frac{{{\epsilon^2}{T^2}}}{2} + \int_{\left| x \right| \ge \epsilon} {2{\rm{d}}{F_{n,k}}\left( x \right)} \\ = \frac{{{\epsilon^2}{T^2}}}{2} + 2P\left( {\left| {{X_{n,k}}} \right| \ge \epsilon} \right),\end{array}$

由1)知存在Ω₀与t无关且P(Ω₀)=1, 使得$\forall \omega \in {\mathit{\Omega} _0}$

$\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} P\left( {\left| {{V_{n,k}}} \right| \ge \epsilon} \right) = 0,$

故有

$\mathop {\lim \;\;\sup }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} {\mathop{\rm Re}\nolimits} \left( {1 - {f_{n,k}}\left( t \right)} \right) \le \frac{{{\epsilon^2}{T^2}}}{2},$

上述收敛性关于$t \in \left[ { - T,T} \right]\;$是一致的, 再让$ \epsilon \to 0$可以得到2).
2) $ \Rightarrow $3).由于

$1 - {f_{n,k}}\left( t \right) = {\mathop{\rm Re}\nolimits} \left( {1 - {f_{n,k}}\left( t \right)} \right) - \int_{ - \infty }^\infty {{\rm{i}}\sin xt{\rm{d}}{F_{n,k}}\left( x \right)} ,$

又因为

$\begin{array}{l}{\left| {\int_{ - \infty }^\infty {{\rm{i}}\sin xt{\rm{d}}{F_{n,k}}\left( x \right)} } \right|^2}\\ \le \int_{ - \infty }^\infty {{{\sin }^2}tx{\rm{d}}{F_{n,k}}\left( x \right)} \\ = \frac{1}{2}\int_{ - \infty }^\infty {\left( {1 - \cos 2tx} \right){\rm{d}}{F_{n,k}}\left( x \right)} \\ = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left( {1 - {f_{n,k}}\left( {2t} \right)} \right),\end{array}$

故由2)可得3).
3)$ \Rightarrow $4)$ \Rightarrow $5).显然.
5)$ \Rightarrow $6).由文献[7]47页(4.8)式知:对任意特征函数f(t)以及与之对应的概率分布函数F(x)均有

$\int_{ - \infty }^\infty {\frac{{{x^2}}}{{1 + {x^2}}}{\rm{d}}F\left( x \right) \le M\left( t \right)\int_0^t {\left( {1 - {\mathop{\rm Re}\nolimits} \left( {f\left( v \right)} \right)} \right){\rm{d}}v,\forall t > 0.} } $

其中, $M\left( t \right) = {\left[ {\mathop {\inf }\limits_{x \in \mathbb{R}} \left( {t\left( {1 - \frac{{\sin \;tx}}{{tx}}} \right)\frac{{1 + {x^2}}}{{{x^2}}}} \right)} \right]^{ - 1}},Re\left( {f\left( v \right)} \right)$为f(v)的实部.
于是

$\begin{array}{l}\mathop {\max }\limits_{1 \le k \le {\xi _n}} \int_{ - \infty }^\infty {\frac{{{x^2}}}{{1 + {x^2}}}{\rm{d}}{F_{n,k}}\left( x \right)} \\ \le \mathop {\max }\limits_{1 \le k \le {\xi _n}} M\left( t \right)\int_0^t {{\mathop{\rm Re}\nolimits} } \left( {1 - {f_{n,k}}\left( v \right)} \right){\rm{d}}v\\ \le M\left( t \right)\int_0^t {\mathop {\max }\limits_{1 \le k{\xi _n}} {\mathop{\rm Re}\nolimits} \left( {1 - {f_{n,k}}\left( v \right)} \right){\rm{d}}v,\forall t > 0,} \end{array}$

由Lebesgue控制收敛定理, 对上式两端取极限, 立得6).
6)$ \Rightarrow $1).任意固定$ \epsilon > 0$, 我们注意到$\frac{{1 + {x^2}}}{{{x^2}}} \le \frac{{1 + {\epsilon^2}}}{{{\epsilon^2}}}$, 当$\left| x \right| \ge \epsilon$, 故

$\begin{array}{l}\mathop {\max }\limits_{1 \le k \le {\xi _n}} P\left( {\left| {{v_{n,k}}} \right| \ge \epsilon} \right)\\ = \mathop {\max }\limits_{1 \le k \le {\xi _n}} \int_{\left| x \right| \ge \epsilon} {\frac{{1 + {x^2}}}{{{x^2}}}\frac{{{x^2}}}{{1 + {x^2}}}{\rm{d}}{F_{n,k}}\left( x \right)} \\ \le \frac{{1 + {\epsilon^2}}}{{{\epsilon^2}}}\mathop {\max }\limits_{1 \le k \le {\xi _n}} \int_{\left| x \right| \ge \epsilon} {\frac{{{x^2}}}{{1 + {x^2}}}{\rm{d}}{F_{n,k}}\left( x \right),} \end{array}$

由6)立得1).
注2.1 ??经典的u.a.n.序列的相应结果见文献[7].
命题2.2 ??设$\left\{ {{\xi _n}:n \ge 1} \right\}$为取值于自然数集的随机变量序列, $\left\{ {{V_{n,k}}:n \ge 1,k \ge 1} \right\}$为关于$\left\{ {{\xi _n}:n \ge 1} \right\}$的u.a.n.序列, 给定$\tau > 0$, 记m_{n, k}为V_{n, k}的中位数, F_{n, k}为V_{n, k}的分布函数, 则:
1) $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} \left| {{m_{n,k}}} \right| = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$;
2) $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} \int_{\left| x \right| xy \tau } {{{\left| x \right|}^r}{\rm{d}}{F_{n,k}}\left( x \right) = 0} \;\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$, 其中常数r>0;
3) $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} \left| {{a_{n,k}}\left( \tau \right)} \right| = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$., 其中${a_{n,k}}\left( \tau \right) = \int_{\left| x \right| \le \tau } {x{\rm{d}}\;{F_{n,k}}\left( x \right)} $, 从而$\left\{ {{V_{n,k}} - {a_{n,k}}\left( \tau \right):k,n \ge 1} \right\}$也为关于$\left\{ {{\xi _n}:n \ge 1} \right\}$的u.a.n.序列;
4) $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{1 \le k \le {\xi _n}} \left| {\exp \left[ { - i{a_{n,k}}\left( \tau \right)t} \right]{f_{n,k}}\left( t \right) - 1} \right| = 0\;\;{\mathbb{\rm{a}}}{\rm{.s}}{\rm{.}}\;{\rm{,}}\;\forall t \in {\rm{R}}$.
证明 ??1)不妨设对于所有的$\omega \in \mathit{\Omega} ,{\xi _n}\left( \omega \right) \to \infty $当n→∞, 任给$ \epsilon > 0,\omega \in \mathit{\Omega} $, 存在正整数$N = N\left( {\epsilon,\omega } \right)$, 使得n>N时

$\mathop {\max }\limits_{1 \le k \le {\xi _n}\left( \omega \right)} P\left( {\left| {{V_{n,k}}} \right| \ge \epsilon} \right) < \frac{1}{2}$

而$P\left( {\left| {{V_{n,k}}} \right| \ge \epsilon} \right) < \frac{1}{2}$意味着$\left| {{m_{n,k}}} \right| \le \epsilon$, 故而

$\mathop {\max }\limits_{1 \le k \le {\xi _n}\left( \omega \right)} \left| {{m_{n,k}}} \right| < \epsilon,n \ge N,\omega \in \mathit{\Omega} ,$

由$\epsilon$的任意性得到1).
2)任给r>0, τ>0以及$0 < \epsilon < \tau $, 注意到

$\begin{array}{l}\mathop {\max }\limits_{1 \le k \le {\xi _n}} \int_{\left| x \right| < \tau } {{{\left| x \right|}^r}{\rm{d}}{F_{n,k}}\left( x \right)} \\ \le \mathop {\max }\limits_{1 \le K \le {\xi _n}} \int_{\left| x \right| < \epsilon} {{{\left| x \right|}^r}{\rm{d}}{F_{n,k}}\left( x \right) + } \\\mathop {\max }\limits_{1 \le k \le {\xi _n}} \int_{\epsilon \le \left| x \right| < \tau } {{{\left| x \right|}^r}{\rm{d}}{F_{n,k}}\left( x \right)} \\ \le {\epsilon^r} + {\tau ^r}\mathop {\max }\limits_{1 \le k \le {\xi _n}} P\left( {\left| {{V_{n,k}}} \right| \ge \epsilon} \right),\end{array}$

再根据$\epsilon$的任意性和$\left\{ {{V_{n,k}}:n \ge 1,k \ge 1} \right\}$是关于$\left\{ {{\xi _n}:n \ge 1} \right\}$的u.a.n.序列这一假设得到结论.
3)在2)中取r=1即可.
4)由3)可知$\left\{ {{V_{n,k}} - {a_{n,k}}\left( \tau \right):k,n \ge 1} \right\}$也是一个关于$\left\{ {{\xi _n}:n \ge 1} \right\}$的u.a.n.序列, 其特征函数恰为$\exp \left\{ { - {\rm{i}}{a_{n,k}}\left( \tau \right)t} \right\}{f_{n,k}}\left( t \right)$, 故由命题2.1中的条件4)可知结论成立.
注2.2 ??该命题推广了经典u.a.n.序列的相应结论, 见文献[7].
定理2.1 ??令${\rm{\mathbb{V}}} = \left\{ {{V_{n,k}}:n \ge 1,k \ge 1} \right\}$为行内独立的二重随机变量序列, 取值于正整数的随机变量序列$\left\{ {{\xi _n}:n \ge 1} \right\}$满足$\mathop {\lim }\limits_{n \to \infty } {\xi _n} = \infty ,{\rm{a}}{\rm{.s}}{\rm{.}}$, 而且$\left\{ {{\xi _n}:n \ge 1} \right\}$与$\left\{ {{V_{n,k}}:n \ge 1,k \ge 1} \right\}$独立, 又设$\left\{ {{V_{n,k}}:n \ge 1,k \ge 1} \right\}$为关于$\left\{ {{\xi _n}:n \ge 1} \right\}$的u.a.n.序列, 记m_{n, k}为V_{n, k}的中位数, F_{n, k}为V_{n, k}的分布函数.若下面2个条件成立:
(a) $\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{\xi _n}} {\int_{\left| x \right| > 1} {{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right) = 0\;{\rm{a}}{\rm{.s}}{\rm{.}}} } $;
(b) $\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{\xi _n}} {\int_{\left| x \right| \le 1} {{x^2}{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right) = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}} } $;
则存在随机变量序列$\left\{ {{V_n},n \ge 1} \right\}$使得

$\sum\limits_{j=1}^{{{\xi }_{n}}}{{{V}_{n,k}}-{{V}_{n}}\xrightarrow{P}0,}\ 当n\to \infty .$

另一方面, 若存在形如h(ξ_n)的随机变量(h是Borel可测函数), 对任意的$\epsilon$>0均有

$\mathop {\lim }\limits_{n \to \infty } \left( {\left| {\sum\limits_{j = 1}^{{\xi _n}} {{V_{n,j}} - h\left( {{\xi _n}} \right)} } \right| \ge \epsilon\left| {{\xi _n}} \right.} \right) = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.,}}$

则(a)与(b)成立.
证明 ??首先证明第1部分.令

$\begin{align} & {{U}_{n}}=\sum\limits_{j=1}^{{{\xi }_{n}}}{{{V}_{n,j}},V_{n,k}^{\left( 1 \right)}={{V}_{n,k}}-{{m}_{n,k}},} \\ & F_{n,k}^{\left( 1 \right)}\left( x \right)=P\left. \left( V_{n,k}^{\left( 1 \right)}\le x \right) \right),V_{n,k}^{\left( 2 \right)}=V_{n,k}^{\left( 1 \right)}{{I}_{\left\{ \left| V_{n,k}^{\left( 1 \right)} \right|\le 1 \right\}}}, \\ & {{V}_{n}}=\sum\limits_{j=1}^{{{\xi }_{n}}}{\left( {{m}_{n,j}}+EV_{n,j}^{\left( 2 \right)} \right),}U_{n}^{\left( 1 \right)}=\sum\limits_{j=1}^{{{\xi }_{n}}}{V_{n,j}^{\left( 1 \right)},} \\ & U_{n}^{\left( 2 \right)}=\sum\limits_{j=1}^{{{\xi }_{n}}}{V_{n,j}^{\left( 2 \right)},{{B}_{n}}=\left\{ U_{n}^{\left( 1 \right)}=U_{n}^{\left( 2 \right)} \right\}.} \\ \end{align}$

由于$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{\xi _n}} {\int_{\left| x \right| > 1} {{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right) = 0} \;\;{\rm{a}}{\rm{.s}}{\rm{.}}} $且$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{\xi _n}} {\int_{\left| x \right| \le 1} {{x^2}{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right) = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}} } $, 故对任意的η>0, 存在${n_0} = {n_0}\left( \eta \right) \in {\rm{\mathbb{N}}}$, 当n≥n₀时, 令${\mathit{\Omega} _0}: = \left\{ {\left( {\int_{\left| x \right| > 1} {{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right)} } \right) \vee \left( {\int_{\left| x \right| \le 1} {{x^2}{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right)} } \right) \le 1} \right\}$满足P(Ω₀)≥1-η, 注意到Ω₀∈σ(ξ_n, n≥1), 故而Ω₀与{V_{n, k}, n≥1, k≥1}独立, 因此

$\begin{array}{l}P\left( {B_n^c \cap {\mathit{\Omega} _0}} \right)\\ = P\left( {\sum\limits_{j = 1}^{{\xi _n}} {V_{n,j}^{\left( 1 \right)} \ne \sum\limits_{j = 1}^{{\xi _n}} {V_{n,k}^{\left( 2 \right)},{\xi _n} = k,{\mathit{\Omega} _0}} } } \right)\\ = \sum\limits_{k = 1}^\infty {P\left( {{\xi _n} = k,{\mathit{\Omega} _0}} \right)P\left( {\sum\limits_{j = 1}^k {V_{n,j}^{\left( 1 \right)} \ne \sum\limits_{j = 1}^k {V_{n,j}^{\left( 2 \right)}} } } \right)} \\ \le \sum\limits_{k = 1}^\infty {P\left( {{\xi _n} = k,{\mathit{\Omega} _0}} \right)\sum\limits_{j = 1}^k {P\left( {V_{n,j}^{\left( 1 \right)} \ne V_{n,j}^{\left( 2 \right)}} \right)} } \\ = \sum\limits_{k = 1}^\infty {P\left( {{\xi _n} = k,{\mathit{\Omega} _0}} \right)\sum\limits_{j = 1}^k {P\left( {\left| {V_{n,j}^{\left( 1 \right)}} \right| > 1} \right)} } \\ = \sum\limits_{k = 1}^\infty {P\left( {{\xi _n} = k,{\mathit{\Omega} _0}} \right)\sum\limits_{j = 1}^k {\int_{\left| x \right| > 1} {{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right)} } }\\= E\left( {\sum\limits_{j = 1}^{{\xi _n}} {\int_{\left| x \right| > 1} {{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right),{\mathit{\Omega} _0}} } } \right), \end{array}$

(1)

再利用控制收敛定理可得, $\mathop {\lim }\limits_{n \to \infty } P\left( {B_n^c \cap {\mathit{\Omega} _0}} \right) = 0.$注意:

$\begin{array}{c}P\left( {\left| {{U_n} - {V_n}} \right| \ge \epsilon,{\mathit{\Omega} _0}} \right)\\ = P\left[ {\left\{ {\left| {{U_n} - {V_n}} \right| \ge \epsilon} \right\} \cap {B_n} \cap {\mathit{\Omega} _0}} \right] + \\P\left[ {\left\{ {\left| {{U_n} - {V_n}} \right| \ge \epsilon} \right\} \cap B_n^c \cap {\mathit{\Omega} _0}} \right],\end{array}$

(2)

前面已证等式右边第2项收敛到0.对第1项利用Chebyshev不等式, 可以得到

$\begin{array}{l}P\left[ {\left\{ {\left| {{U_n} - {V_n}} \right| \ge \epsilon} \right\} \cap {B_n} \cap {\mathit{\Omega} _0}} \right]\\ = P\left[ {\left\{ {\left| {\sum\limits_{j = 1}^{{\xi _n}} {\left( {V_{n,k}^{\left( 2 \right)} - EV_{n,k}^{\left( 2 \right)}} \right)} } \right| \ge \epsilon} \right\} \cap {B_n} \cap {\mathit{\Omega} _0}} \right]\\ \le \sum\limits_{k = 1}^\infty {P\left( {{\xi _n} = k,{\mathit{\Omega} _0}} \right)P\left( {\left| {\sum\limits_{j = 1}^k {\left( {V_{n,k}^{\left( 2 \right)} - EV_{n,j}^{\left( 2 \right)}} \right)} } \right| \ge \epsilon} \right)} \\ \le \sum\limits_{k = 1}^\infty {P\left( {{\xi _n} = k,{\mathit{\Omega} _0}} \right)\frac{1}{{{\epsilon^2}}}\sum\limits_{j = 1}^k {{\rm{Var}}\left( {V_{n,k}^{\left( 2 \right)}} \right)} } \\ = \frac{1}{{{\epsilon^2}}}E\left( {\sum\limits_{k = 1}^{{\xi _n}} {{\rm{Var}}\left( {V_{n,j}^{\left( 2 \right)}} \right),{\mathit{\Omega} _0}} } \right)\\ = \frac{1}{{{\epsilon^2}}}E\left( {\sum\limits_{j = 1}^{{\xi _n}} {\int_{\left| x \right| \le 1} {{x^2}{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right),{\mathit{\Omega} _0}} } } \right),\end{array}$

因为在Ω₀上$\left| {\sum\limits_{j = 1}^{{\xi _n}} {\int_{\left| x \right| \le 1} {{x^2}{\rm{d}}{F_{n,j}}\left( {x + {m_{n,j}}} \right)} } } \right| \le 1$, 所以由控制收敛定理知(2)式右端第1项亦收敛于0.总之

$\mathop {\lim }\limits_{n \to \infty } P\left( {\left| {{U_n} - {V_n}} \right| \ge \epsilon} \right) = 0.$

因此

$\sum\limits_{j=1}^{{{\xi }_{n}}}{{{V}_{n,j}}-{{V}_{n}}}\xrightarrow{P}0,n\to \infty .$

另一方面, 对于每个$\epsilon$>0, 由条件

$P\left( \left| \sum\limits_{j=1}^{{{\xi }_{n}}}{{{V}_{n,j}}-h\left( {{\xi }_{n}} \right)} \right|\ge \epsilon \left| {{\xi }_{n}} \right. \right)\to 0\ \ \text{a}\text{.s}\text{.}$

可得

$P\left( \sum\limits_{j=1}^{{{\xi }_{n}}}{{{V}_{n,j}}-h\left( {{\xi }_{n}} \right)\le x\left| {{\xi }_{n}} \right.} \right)\text{a}\text{.s}\text{.}\left\{ \begin{align} & 1,x>0, \\ & 0, < 0. \\ \end{align} \right.$

故任取T>0, 下式对u∈I=[-T, T]一致成立

$\mathop {\lim }\limits_{n \to \infty ]} E{\mkern 1mu} \left( {\exp \left( {{\rm{i}}u\left( {\sum\limits_{j = 1}^{{\xi _n}} {{V_{n,j}} - h\left( {{\xi _n}} \right)} } \right)} \right)\left| {{\xi _n}} \right.} \right)\left( \omega \right) = 1\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$

而

$\begin{align} & E\left( \exp \left( \text{i}u\left( \sum\limits_{j=1}^{{{\xi }_{n}}}{Vn,j-h\left( {{\xi }_{n}} \right)} \right) \right)\left| {{\xi }_{n}} \right. \right) \\ & \ \ \ =\exp \left\{ -\text{i}uh\left( {{\xi }_{n}} \right) \right\}\prod\limits_{j=1}^{{{\xi }_{n}}}{{{f}_{n,j}}\left( u \right)}, \\ \end{align}$

其中，f_{n, j}(u)为V_{n, j}的特征函数, 于是对于a.s.ω, $\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{\xi _n}} {1 - {{\left| {{f_{n,j}}\left( u \right)} \right|}^2} = 1} $在I上一致成立.对上式两端取对数可得: $\forall u \in I$,

$\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{j=1}^{{{\xi }_{n}}}{\log }\left( 1-\left( 1-{{\left| {{f}_{n,j}}\left( u \right) \right|}^{2}} \right) \right)=0\ \ \text{a}\text{.s}\text{.}$

(3)

根据命题2.1, 对于a.s.ω, $\mathop {\max }\limits_{1 \le j \le {\xi _n}} \left| {1 - {{\left| {{f_{n,j}}\left( u \right)} \right|}^2}} \right|$在I上一致地收敛到零(当n→∞), 利用不等式$ - \log \left( {1 - x} \right) \ge x\left( {0 < x < 1} \right)$可得:对a.s.ω,

$\begin{align} & \text{-}\sum\limits_{j=1}^{{{\xi }_{n}}}{\log \left( 1-\left( 1-{{\left| {{f}_{n,j}}\left( u \right) \right|}^{2}} \right) \right)\ge \sum\limits_{j=1}^{{{\xi }_{n}}}{\left( 1-{{\left| {{f}_{n,j}}\left( u \right) \right|}^{2}} \right)}} \\ & \ \ \ge 0,\forall u\in I,n\ge {{n}_{0}}={{n}_{0}}\left( \omega \right), \\ \end{align}$

利用式(3)有:对于a.s.ω, $\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{\xi _n}} {\left( {1 - {{\left| {{f_{n,j}}\left( u \right)} \right|}^2}} \right) = 0\left( {对u \in I一致} \right)} $.取I=[-1, 1], 由一致收敛性有

$\underset{n\to \infty }{\mathop{\lim }}\,\int_{-1}^{1}{\sum\limits_{j=1}^{{{\xi }_{n}}}{\left( 1-{{\left| {{f}_{n,j}}\left( u \right) \right|}^{2}} \right)\text{d}u=0\ \ \ \text{a}\text{.s}\text{.}}}$

(4)

令$\left\{ {{{\bar V}_{n,k}},n \ge 1,k \ge 1} \right\}$为{V_{n, k}}的一个独立复制.令G_{n, j}(x)为$\left( {{{\bar V}_{n,j}} - {V_{n,j}}} \right)$的分布, 则${\left| {{f_{n,j}}\left( u \right)} \right|^2}为{G_{n,j}}\left( x \right)$对应的特征函数, 由Fubini定理得到

$\int_{-1}^{1}{\left( 1-{{\left| {{f}_{n,j}}\left( u \right) \right|}^{2}} \right)}\text{d}u=2\int_{-\infty }^{\infty }{\left( 1-\frac{\sin \ x}{x} \right)\text{d}\ {{G}_{n,j}}\left( x \right).}$

接下来分析一些简单的估计, 首先存在正常数K₁, 使得|x|>1时, $1 - \frac{{\sin \;x}}{x} \ge {K_1} > 0$.对于|x|≤1存在正常数K₂使得

$1-\frac{\sin x}{x}=1-\frac{1}{x}\left( x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}-+\cdots \right)\ge {{K}_{2}}{{x}^{2}}.$

令K₃=min{K₁, K₂}>0, 令$\psi \left( x \right) = {x^2}{I_{\left\{ {\left| x \right| \le 1} \right\}}} + {I_{\left\{ {\left| x \right| > 1} \right\}}}$, 由文献[2]中169页引理1, $E\psi \left( {{{\bar V}_{n,j}} - {V_{n,j}}} \right) \ge \frac{1}{2}E\psi \left( {{{\bar V}_{n,j}} - {m_{n,j}}} \right)$则

$\begin{align} & \int\limits_{-1}^{1}{\sum\limits_{j=1}^{{{\xi }_{n}}}{\left( 1-{{\left| {{f}_{n,j}}\left( u \right) \right|}^{2}} \right)\text{d}u}} \\ & \ge {{K}_{3}}\left\{ \sum\limits_{j=1}^{{{\xi }_{n}}}{\int_{\left| x \right|\le 1}{{{x}^{2}}\text{d}{{G}_{n,j}}\left( x \right)+\sum\limits_{j=1}^{{{\xi }_{n}}}{\int_{\left| x \right|>1}{\text{d}{{G}_{n,j}}\left( x \right)}}}} \right\} \\ & ={{K}_{3}}\sum\limits_{j=1}^{{{\xi }_{n}}}{E\psi \left( {{{\bar{V}}}_{n,j}}-{{V}_{n,j}} \right)\ge \frac{{{K}_{3}}}{2}\sum\limits_{j=1}^{{{\xi }_{n}}}{E\psi \left( {{{\bar{V}}}_{n,j}}-{{m}_{n,j}} \right)}} \\ & =\frac{{{K}_{3}}}{2}\sum\limits_{j=1}^{{{\xi }_{n}}}{\left\{ \int_{\left| x \right|>1}{\text{d}{{F}_{n,j}}\left( x+{{m}_{n,j}} \right)}+ \right.} \\ & \left. \int_{\left| x \right|\le 1}{{{x}^{2}}\text{d}{{F}_{n,j}}\left( x+{{m}_{n,j}} \right)} \right\}\ge 0. \\ \end{align}$

由于括号内两式均非负, 对两边取极限得到(a)和(b).
下面讨论另一种情形.仍然沿用前面的符号.设{Z_n:n≥0}为定义1.1所定义的变化环境中的上临界的分枝过程, 记${Y_{n,k}} = \frac{{{X_{n,k}} - {\mu _n}}}{{f\left( {{Z_n}} \right)}}:n \ge 1,k \ge 1.$. f是单调上升函数且$\mathop {\lim }\limits_{x \to \infty } \left( x \right) = \infty $.记ν_{n, k}为Y_{n, k}关于Z_n的条件中位数(即$P\left( {{Y_{n,k}} \ge {v_{n,k}}\left| {{Z_n}} \right.} \right) \ge \frac{1}{2},P\left( {{Y_{n,k}} \le {v_{n,k}}\left| {{Z_n}} \right.} \right) \ge \frac{1}{2}$), F_{n, k}为Y_{n, k}关于Z_n的条件分布函数(即${F_{n,k}}\left( x \right) = P\left( {{Y_{n,k}} \le x\left| {{Z_n}} \right.} \right)$).
定理2.2 ??假设Z_n→∞a.s., 当n→∞, 若下面两个条件成立:
(a′)$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{Z_n}} {\int_{\left| x \right| > 1} {{\rm{d}}{F_{n,j}}\left( {x + {v_{n,j}}} \right) = 0} \;{\rm{a}}{\rm{.s}}{\rm{.}}} $;
(b′) $\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 1}^{{Z_n}} {\int_{\left| x \right| \le 1} {{x^2}{\rm{d}}{F_{n,j}}\left( {x + {v_{n,j}}} \right) = 0} \;{\rm{a}}{\rm{.s}}{\rm{.}}} $;
则存在随机变量序列$\left\{ {{H_n}:n \ge 1} \right\}$使得:对于任意的$\epsilon > 0$,

$\mathop {\lim }\limits_{n \to \infty } P\left( {\left| {\sum\limits_{j = 1}^{{Z_n}} {{Y_{n,j}} - {H_n}} } \right| \ge \epsilon\left| {{Z_n}} \right.} \right) = 0\;\;\;{\rm{a}}.{\rm{s}}.,$

进而

$\sum\limits_{j=1}^{{{Z}_{n}}}{{{Y}_{n,j}}-{{H}_{n}}\xrightarrow{P}0,}为n\to \infty .$

证明 ??首先令

$\begin{align} & Y_{n,k}^{\left( 1 \right)}={{Y}_{n,k}}-{{v}_{n,k}},F_{n,k}^{\left( 1 \right)}\left( x \right)=P\left( Y_{n,k}^{(1)}\le x\left| {{Z}_{n}} \right. \right), \\ & Y_{n,k}^{\left( 2 \right)}=Y_{n,k}^{\left( 1 \right)}{{I}_{\left\{ \left| Y_{n,k}^{\left( 1 \right)} \right|\le 1 \right\}}},{{H}_{n}}=\sum\limits_{j=1}^{{{Z}_{n}}}{\left( {{v}_{n,j}}+EY_{n,j}^{\left( 2 \right)} \right),} \\ & {{Y}_{n}}=\sum\limits_{j=1}^{{{Z}_{n}}}{{{Y}_{n,j}},Y_{n}^{\left( 1 \right)}=\sum\limits_{j=1}^{{{Z}_{n}}}{Y_{n,j}^{\left( 1 \right)}},} \\ & Y_{n}^{\left( 2 \right)}=\sum\limits_{j=1}^{{{Z}_{n}}}{Y_{n,j}^{\left( 2 \right)},{{C}_{n}}=\left\{ Y_{n}^{\left( 1 \right)}=Y_{n}^{\left( 2 \right)} \right\}.} \\ \end{align}$

注意到

$\begin{align} & P\left( C_{n}^{c}\left| {{Z}_{n}} \right. \right)=P\left( Y_{n}^{\left( 1 \right)}\ne Y_{n}^{\left( 2 \right)}\left| {{Z}_{n}} \right. \right), \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =P\left( \sum\limits_{j=1}^{{{Z}_{n}}}{Y_{n,j}^{\left( 1 \right)}\ne \sum\limits_{j=1}^{{{Z}_{n}}}{Y_{n,j}^{\left( 2 \right)}\left| {{Z}_{n}} \right.}} \right), \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le \sum\limits_{k=1}^{{{Z}_{n}}}{P\left( Y_{n,j}^{\left( 1 \right)}\ne Y_{n,j}^{\left( 2 \right)}\left| {{Z}_{n}} \right. \right),} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\sum\limits_{j=1}^{{{Z}_{n}}}{\int_{\left| x \right|>1}{\text{d}{{F}_{n,j}}\left( x+{{v}_{n,j}} \right),}} \\ \end{align}$

根据(a′)知, $\mathop {\lim }\limits_{n \to \infty } P\left( {C_n^c\left| {{Z_n}} \right.} \right) = 0\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$.又有

$\begin{align} & P\left( \left\{ \left| \sum\limits_{k=1}^{{{Z}_{n}}}{{{Y}_{n,j}}-{{H}_{n}}} \right|\ge \epsilon \right\},{{C}_{n}}\left| {{Z}_{n}} \right. \right) \\ & =P\left( \left| \sum\limits_{j=1}^{{{Z}_{n}}}{\left( Y_{n,j}^{\left( 2 \right)}-EY_{n,j}^{\left( 2 \right)} \right)\left| \ge \epsilon,{{C}_{n}} \right.} \right|{{Z}_{n}} \right) \\ & \le P\left( \left| \sum\limits_{j=1}^{{{Z}_{n}}}{\left( Y_{n,j}^{\left( 2 \right)}-EY_{n,j}^{\left( 2 \right)} \right)} \right|\ge \epsilon\left| {{Z}_{n}} \right. \right) \\ & \le \sum\limits_{k=1}^{{{Z}_{n}}}{P\left( \left| Y_{n,j}^{\left( 2 \right)}-EY_{n,j}^{\left( 2 \right)} \right|\ge \epsilon\left| {{Z}_{n}} \right. \right)} \\ & =\sum\limits_{k=1}^{\infty }{\frac{1}{{{\epsilon}^{2}}}\text{Var}\left( Y_{n,j}^{\left( 2 \right)}\left| {{Z}_{n}} \right. \right)} \\ & \le \sum\limits_{k=1}^{\infty }{\frac{1}{{{\epsilon}^{2}}}E\left( {{\left| Y_{n,j}^{\left( 2 \right)} \right|}^{2}}\left| {{Z}_{n}} \right. \right)} \\ & =\frac{1}{{{\epsilon}^{2}}}\sum\limits_{k=1}^{\infty }{\int_{\left| x \right|\le 1}{{{x}^{2}}\text{d}{{F}_{n,j}}\left( x+{{v}_{n,j}} \right),}} \\ \end{align}$

由(b′), $\mathop {\lim }\limits_{n \to \infty } P\left( {\left| {\sum\limits_{j = 1}^{{Z_n}} {{Y_{n,j}} - {H_n}} } \right| \ge \epsilon,{C_n}\left| {{Z_n}} \right.} \right) = 0\;\;{\rm{a}}{\rm{.s}}$总之

$\mathop {\lim }\limits_{n \to \infty } {\mkern 1mu} P\left( {\left| {\sum\limits_{j = 1}^{{Z_n}} {{Y_{n,j}} - {H_n}} } \right| \ge \epsilon\left| {{Z_n}} \right.} \right) = 0\;\;\;{\rm{a}}{\rm{.s}}{\rm{.}}$

利用控制收敛定理得

$\sum\limits_{j=1}^{{{Z}_{n}}}{{{Y}_{n,j}}-{{H}_{n}}\xrightarrow{P}0,当n\to \infty .}$

参考文献

[1]	Gendenko B V, Kolmogorov A N. Limit distributions for sums of independent random variables[M]. Chung K L, trans. Cambridge:Addison-Wesley, 1954.
[2]	Tucker H G. A graduate course in probability[M].New York and London: Academic Press, 1967.
[3]	胡迪鹤. 不变原理及其在分枝过程中的应用[J].北京大学学报:自然科学版, 1964(1):1–27.
[4]	Heyde C C. Some central limit analogues for supercritical Galton-Watson processes[J].Appl Prob, 1971, 8(1):52–59.
[5]	高振龙.随机环境中Galton-Watson分枝过程的极限定理[D].北京:中国科学院研究生院, 2011.
[6]	Gao Z L, Hu X Y. Limit theorems for a Galton-Watson process in the i.i.d. random environment[J]. Acta Mathematica Scientia, 2012, 32(3):1193-1205.
[7]	胡迪鹤. 分析概率论[M].北京: 科学出版社, 1997.
[8]	Athreya K B, Ney P E. Branching processes[M].Berlin, Heidelberg, New York: Springer-Verlag, 1972.