在一维线性结构关系EV(errors-in-variables)模型^[1]中, 如果参数$a, b$是实变量t的有界连续函数$a(t), b(t), t\in(0, 1), (b(t)\neq0)$, 可得到下列变系数一维线性结构关系的EV模型

$\left\{ \begin{array}{*{35}{l}} y=a(t)+b(t)x, \\ Y=y+\epsilon , \\ X=x+u, \\ \end{array} \right.$

其中$X, Y$是随机变量, $(\epsilon, u)$是测量误差, t是一实变量, 可以是温度、时间等, 假定t在一个闭区间上变化, 可通过变换使得$t\in[0,1]$.

目前关于变系数EV模型的讨论还处在起步阶段, 但是也取得了一些成果, 如欧阳^[2]初步研究了这类模型, 他利用加权正交回归最小二乘法给出了该模型的一维线性结构的参数估计, 并得到了该估计的强、弱相合性; 崔^[3]给出了变系数线性EV模型参数的调整加权最小二乘估计及其渐近性质; 方和胡^[4]讨论了核实数据下非线性EV模型中经验似然降维推断等.而本文则研究变系数EV模型的ND样本加权和的相合性问题, 为此先讨论EV模型的权函数问题.

设$t_{0}\in(0, 1)$, 要对$t_{0}$处的$a(t_{0}), b(t_{0})$进行参数估计.然而不可能在$t_{0}$处作$n$次观测, 只能在$t_{0}$处附近作$n$次观测.设$t_{1}, t_{2}, \cdots, t_{n}$是$[0,1]$上$n$个设计点, 满足

$0\leq t_{1}\leq t_{2}\leq, \cdots, \leq t_{n}\leq 1.$

对每个点$t_{i}$处$(Y, X)$作观测, 得到$n$组观测值$(Y_{i}, t_{i}, X_{i}) (i=1, 2, \cdots, n)$.当利用这$n$组观测值来估计$t_{0}$处的参数$a(t_{0}), b(t_{0})$时, 此时应该注意到$t_{i}$处的观测值$(Y_{i}, t_{i}, X_{i}) (i=1, 2, \cdots, n)$相对于$t_{0}$来说它们的重要程度并不一样, 这种重要程度可用实变量$t_{i}$的权函数$W_{ni}(t_{0})$来度量.下面给出权函数的定义.

设$(Y_{i}, t_{i}, X_{i}) (i=1, 2, \cdots, n)$是取自母体$(Y, X)$的样本, $t_{1}, t_{2}, \cdots, t_{n}$是$[0,1]$上的$n$个设计点, $t_{0}$是$(0, 1)$内的某一个点, 实变量$t_{1}, t_{2}, \cdots, t_{n}$的函数$W_{ni}(t_{0})=W_{ni}(t_{0}, t_{1}, t_{2}, \cdots, t_{n})$ $(i=1, 2, \cdots, n)$称为实变量权函数(简称为权函数), 如果它满足

$(1) W_{ni}(t_{0})＞0 (i=1, 2, \cdots, n);$

$(2) \sum\limits_{i=1}^{n}W_{ni}(t_{0})=1.$

选定一维概率密度函数$k(\cdot)$及窗宽$h_{n}\in (0, 1/2), h_{n}\to 0 (n\to \infty)$, 则有

${{W}_{ni}}({{t}_{0}})=\frac{k[({{t}_{i}}-{{t}_{0}})/{{h}_{n}}]}{\sum\limits_{i=1}^{n}{k}[({{t}_{i}}-{{t}_{0}})/{{h}_{n}}]}$

为权函数, 称之为核权函数.

在文[5]中提出了这样一个问题:在何种条件下, 当$n\to \infty $时有

$\sum\limits_{i=1}^{n}{{{W}_{ni}}}({{x}_{0}}){{Y}_{i}}\to E(Y|X={{x}_{0}})\,\text{a}.\text{s}..$

欧阳^[6]研究了上式中的权函数$W_{ni}(x_{0})$为实变量核权函数$W_{ni}(t_{0})$时, 独立随机变量序列$\{Y_{i}\}_{i=1}^{n}$加权和$\sum\limits_{i=1}^{n}W_{ni}(t_{0})Y_{i}$的相合性, 付和吴^[7]研究了NA同分布序列$\{Y_{i}\}_{i=1}^{n}$加权和的相合性, 而有关此问题的讨论的其他文献目前的报道较少.本文在此基础上进行推广, 研究变系数EV模型的ND样本$\{Y_{i}\}_{i=1}^{n}$加权和的相合性, 获得了与独立随机变量样本加权和相同的结论.为此, 需给出ND序列的概念.

定义^[8]  称随机变量$X_{1}, X_{2}, \cdots, X_{n}, n\geq2$是ND (Negatively Dependent)的, 若对任意$x_{1}, x_{2}, \cdots, x_{n}\in R, $有

$P\left( \bigcap\limits_{j=1}^{n}{({{X}_{j}}\le {{x}_{j}})} \right)\le \prod\limits_{j=1}^{n}{P}({{X}_{j}}\le {{x}_{j}}),P\left( \bigcap\limits_{j=1}^{n}{({{X}_{j}}>{{x}_{j}})} \right)\le \prod\limits_{j=1}^{n}{P}({{X}_{j}}>{{x}_{j}}).$

如果$\forall n\geq 2$, $X_{1}, X_{2}, \cdots, X_{n}$都是ND的, 则称随机变量列$\{X_{n}, n\geq1\}$是ND列.

文献[9]举例说明了NA序列一定是ND序列, 但ND序列不一定是NA序列, 这说明ND序列是比NA序列更弱、更广泛的一种随机变量序列.因此, 对ND列的研究在理论和实践中都是十分有意义的.自从1993年Bozorgnia等^[8]提出ND相依概念以来, 已经引起了越来越多的学者的关注, 也取得了许多的研究成果, 例如文献[8-10]等.

为了得出本文的主要结论, 本节先给出一些相关的引理.

引理1 ^[8] 设$\{X_{n}, n\geq1\}$是ND的, $\forall m\geq2, A_{1}, A_{2}, \cdots, A_{m}$是集合$\{1, 2, \cdots, n\}$的两两不交的非空子集.如果$f_{i}, i=1, 2, \cdots, m$是对每个变元都非降(或都非升)的函数, 则$f_{1}(X_{j}, j\in A_{1}), f_{2}(X_{j}, j\in A_{2}), \cdots, f_{m}(X_{j}, j\in A_{m})$仍是ND的.

引理2 ^[8] 设随机变量$\{X_{n}, n\geq1\}$是ND列, 则

$E\left(\prod\limits_{i=1}^{n}X_{i}\right)\leq \prod\limits_{i=1}^{n}EX_{i}.$

特别地, 设随机变量$\{X_{n}, n\geq1\}$是ND列, $t_{1}, t_{2}, \cdots, t_{n}$都是非正或者都是非负的实数, 则有

$E\left[\text{exp}\left(\sum\limits_{i=1}^{n}t_{i}X_{i}\right)\right]\leq \prod\limits_{i=1}^{n}E\left[\text{exp}(t_{i}X_{i})\right].$

引理3 ^[9] (Bernstein不等式)设随机变量$\{X_{n}, n\geq1\}$是ND列, $EX_{i}=0, |X_{i}|\leq b_{i}$, a.s., $(i=1, 2, \cdots, n), t＞0$为实数, 且满足$t\mathop {\max }\limits_{1 \le i \le n} {b_i} \le 1$, 则$\forall\epsilon>0$, 有

$P\left(\left|\sum\limits_{i=1}^{n}X_{i}\right|>\epsilon\right)\leq2\text{exp}\left\{-t\epsilon+t^{2}\sum\limits_{i=1}^{n}EX_{i}^{2}\right\}.$

引理4 设$X_{1}, X_{2}, \cdots, X_{n}$是ND的, $EX_{i}=0$, 且

$\mathop {\max }\limits_{1 \le i \le n} |{X_i}| = o\left( {{{({\rm{log}}n)}^{ - 1}}} \right),\sum\limits_{i = 1}^n E X_i^2 = o\left( {{{({\rm{log}}n)}^{ - 1}}} \right),$

则$\forall \epsilon>0$及充分大的$n$, 有$P\left(\left|\sum\limits_{i=1}^{n}X_{i}\right|>\epsilon\right)\leq 2n^{-2}.$

证    由条件, $\forall \epsilon>0$及充分大的$n$知

$\mathop {\max }\limits_{1 \le i \le n} |{X_i}| \le {\epsilon^{2}}{(16{\rm{log}}\,n)^{ - 1}},\sum\limits_{i = 1}^n E X_i^2 \le {\epsilon^{2}}{(16{\rm{log}}n)^{ - 1}}.$

取$t=4\epsilon^{-1}\text{log} n＞0$, 则t满足引理3的要求, 因此

$P\left(\left|\sum\limits_{i=1}^{n}X_{i}\right|>\epsilon\right)\leq 2 \text{exp}\{-4 \text{log} n+\text{log} n\}\leq 2 \text{exp}\{-2 \text{log} n\}=2n^{-2}.$

定理1     设$\{Y, Y_{i}, i\geq1\}$为同分布的ND样本序列, 且存在$M＞0$, 使得$\text{Var}(Y)\leq M$, 若对任意实变量核权函数$\{W_{ni}(t_{0})\}_{i=1}^{n}$, 存在正数$C$, 使得

$\mathop {\max }\limits_{1 \le i \le n} {W_{ni}}({t_0}) \le \frac{{C{\rm{log}}n}}{{n{h_n}}},$

则有

(1) 当$\frac{\text{log} n}{nh_{n}}\rightarrow 0 (n\rightarrow\infty)$时,

$\sum\limits_{i=1}^{n}{{{W}_{ni}}}({{t}_{0}}){{Y}_{i}}\xrightarrow{p}EY;$

(3.1)

(2) 当$\frac{\text{log}^{2} n}{\sqrt{n}h_{n}}\rightarrow 0 (n\rightarrow\infty)$时,

$\sum\limits_{i=1}^{n}{{{W}_{ni}}}({{t}_{0}}){{Y}_{i}}\xrightarrow{\text{a}.\text{s}.}EY.$

(3.2)

注    由于ND序列是比独立序列和NA序列更弱、更广泛的一种随机变量序列, 所以该定理在较弱的条件下, 得到了与独立情形下相同的结论.

证    由于式(3.1) 和(3.2) 可以转化为

$\sum\limits_{i=1}^{n}W_{ni}(t_{0})(Y_{i}-EY_{i})\xrightarrow{p} 0, \sum\limits_{i=1}^{n}W_{ni}(t_{0})(Y_{i}-EY_{i})\xrightarrow{\text{a}.\text{s}.} 0.$

而$E(Y_{i}-EY_{i})=0, \text{Var}(Y_{i}-EY_{i})=\text{Var}(Y_{i})\leq M, i=1, 2, \cdots, n$.因此不失一般性, 不妨假设$EY=0$.

先证结论(1) 成立, 因为$E\left(\sum\limits_{i=1}^{n}W_{ni}(t_{0})Y_{i}\right)=0$, 为证式(3.1) 成立, 由Markov条件, 只需证明$\mathop {\lim }\limits_{n \to \infty } {\rm{Var}}\left( {\sum\limits_{i = 1}^n {{W_{ni}}} ({t_0}){Y_i}} \right) = 0$即可.

事实上

${\rm{Var}}\left( {\sum\limits_{i = 1}^n {{W_{ni}}} ({t_0}){Y_i}} \right) \le \sum\limits_{i = 1}^n {W_{ni}^2} ({t_0}){\rm{Var}}({Y_i}) \le M\left( {\mathop {\max }\limits_{1 \le i \le n} {W_{ni}}({t_0})} \right) \le \frac{{MC{\rm{log}}\,n}}{{n{h_n}}}.$

故由条件$\frac{\text{log} n}{nh_{n}}\rightarrow 0 (n\rightarrow\infty)$可知$\mathop {\lim }\limits_{n \to \infty } {\rm{Var}}\left( {\sum\limits_{i = 1}^n {{W_{ni}}} ({t_0}){Y_i}} \right) = 0$, 从而(1) 得证.

下证结论(2) 成立. $\forall i\leq n$, 令

$Y_{i}^{(1)}=-\sqrt{n}I_{(Y_{i}＜-\sqrt{n})}+Y_{i}I_{(|Y_{i}|\leq \sqrt{n})}+\sqrt{n}I_{(Y_{i}＞\sqrt{n})}, \\ Y_{i}^{(2)}=Y_{i}-Y_{i}^{(1)}=(Y_{i}+\sqrt{n})I_{(Y_{i}＜-\sqrt{n})}+(Y_{i}-\sqrt{n})I_{(Y_{i}＞\sqrt{n})}, $

则$Y_{i}=Y_{i}^{(1)}+Y_{i}^{(2)}$, 由$EY=0$, 得$EY_{i}^{(1)}=-EY_{i}^{(2)}$, 故得$Y_{i}=(Y_{i}^{(1)}-EY_{i}^{(1)})+(Y_{i}^{(2)}-EY_{i}^{(2)}), $从而

$\sum\limits_{i=1}^{n}W_{ni}(t_{0})Y_{i}=\sum\limits_{i=1}^{n}W_{ni}(t_{0})(Y_{i}^{(1)}-EY_{i}^{(1)})+\sum\limits_{i=1}^{n}W_{ni}(t_{0})Y_{i}(Y_{i}^{(2)}-EY_{i}^{(2)}).$

(3.3)

要证式(3.2) 成立, 只需证下面两式成立即可

$\sum\limits_{i=1}^{n}{{{W}_{ni}}}({{t}_{0}})\left( Y_{i}^{(1)}-EY_{i}^{(1)} \right)\xrightarrow{\text{a}.\text{s}.}0;$

(3.4)

$\sum\limits_{i=1}^{n}{{{W}_{ni}}}({{t}_{0}})\left( Y_{i}^{(2)}-EY_{i}^{(2)} \right)\xrightarrow{\text{a}.\text{s}.}0.$

(3.5)

先证式(3.4) 成立, 记${X_{ni}} \buildrel\textstyle.\over= {W_{ni}}({t_0})(Y_i^{(1)} - EY_i^{(1)}),{b_n} \buildrel\textstyle.\over= \mathop {\max }\limits_{1 \le i \le n} {W_{ni}}({t_0})$.由$W_{ni}(t_{0})>0$和$Y_{i}^{(1)}$的定义, 结合引理1知$\{X_{ni}\}_{i=1}^{n}$仍是零均值同分布ND序列.

由于

$\begin{align} &\left| Y_{i}^{(1)}-EY_{i}^{(1)} \right|=\left| -\sqrt{n}{{I}_{({{Y}_{i}}＜-\sqrt{n})}}+{{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}}+\sqrt{n}{{I}_{({{Y}_{i}}＞\sqrt{n})}} \right. \\ &\left. \quad \quad \quad \quad \quad \quad \quad -E\left( -\sqrt{n}{{I}_{({{Y}_{i}}＜-\sqrt{n})}}+{{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}}+\sqrt{n}{{I}_{({{Y}_{i}}＞\sqrt{n})}} \right) \right| \\ &\quad \quad \quad \quad \quad \le \sqrt{n}\left| {{I}_{({{Y}_{i}}＜-\sqrt{n})}}-P({{Y}_{i}}＜-\sqrt{n}) \right|+\sqrt{n}\left| {{I}_{({{Y}_{i}}＞\sqrt{n})}}-P({{Y}_{i}}＞\sqrt{n}) \right| \\ &\quad \quad \quad \quad \quad \quad \quad +\left| {{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}}-E{{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}} \right| \\ &\quad \quad \quad \quad \quad \le \sqrt{n}\left| {{I}_{({{Y}_{i}}＜-\sqrt{n})}} \right|+\sqrt{n}\left| {{I}_{({{Y}_{i}}＞\sqrt{n})}} \right|+\left| {{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}}-E{{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}} \right| \\ &\quad \quad \quad \quad \quad \le 2\sqrt{n}+\left| {{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}} \right|+\left| E{{Y}_{i}}{{I}_{(|{{Y}_{i}}|\le \sqrt{n})}} \right|\le 4\sqrt{n}, \\ \end{align}$

(3.6)

从而$\mathop {\max }\limits_{1 \le i \le n} \left| {{X_{ni}}} \right| = \mathop {\max }\limits_{1 \le i \le n} {W_{ni}}({t_0})\left| {Y_i^{(1)} - EY_i^{(1)}} \right| \le 4{b_n}\sqrt n .$又因

$\begin{array}{l} \sum\limits_{i = 1}^n {{\rm{Var}}} \left( {{X_{ni}}} \right) = \sum\limits_{i = 1}^n {{\rm{Var}}} \left[ {{W_{ni}}({t_0})\left( {Y_i^{(1)} - EY_i^{(1)}} \right)} \right] = \sum\limits_{i = 1}^n {W_{ni}^2} ({t_0}){\rm{Var}}\left( {Y_i^{(1)}} \right)\\ \quad \quad \quad \quad \quad \le \sum\limits_{i = 1}^n {{{\left[ {\mathop {\max }\limits_{1 \le i \le n} {W_{ni}}({t_0})} \right]}^2}} {\rm{Var}}\left( {{Y_i}} \right) \le nMb_n^2, \end{array}$

所以由$\frac{\text{log}^{2} n}{\sqrt{n}h_{n}}\rightarrow 0 (n\rightarrow\infty)$知

$\begin{array}{l} \frac{{\mathop {\max }\limits_{1 \le i \le n} \left| {{X_{ni}}} \right|}}{{{\rm{log}}n}} \le \frac{{4{b_n}\sqrt n }}{{{\rm{log}}n}} \le \frac{{4\sqrt n }}{{{\rm{log}}n}}\frac{{C{\rm{log}}n}}{{n{h_n}}} = \frac{{4C}}{{\sqrt n {h_n}}} \to 0,n \to \infty ;\\ \frac{{\sum\limits_{i = 1}^n E X_{ni}^2}}{{{\rm{log}}n}} \le \frac{{\sum\limits_{i = 1}^n {{\rm{Var}}} \left( {{X_{ni}}} \right)}}{{{\rm{log}}n}} \le \frac{{nMb_n^2}}{{{\rm{log}}n}} \le \frac{{nM}}{{{\rm{log}}n}}\frac{{{C^2}{\rm{lo}}{{\rm{g}}^2}n}}{{{n^2}h_n^2}}\\ \quad \quad \quad \quad = \frac{{{C^2}M}}{{{\rm{lo}}{{\rm{g}}^3}n}}{\left( {\frac{{{\rm{lo}}{{\rm{g}}^2}n}}{{\sqrt n {h_n}}}} \right)^2} \to 0,n \to \infty . \end{array}$

由上面两式可得$\mathop {\max }\limits_{1 \le i \le n} \left| {{X_{ni}}} \right| = o\left( {{{({\rm{log}}n)}^{ - 1}}} \right),\sum\limits_{i = 1}^n E X_{ni}^2 = o\left( {{{({\rm{log}}n)}^{ - 1}}} \right).$

结合引理4知, $\forall \epsilon＞0$及充分大的$n$, 有

$P\left(\left|\sum\limits_{i=1}^{n}X_{ni}\right|>\epsilon\right)\leq 2n^{-2}, $

故

$\sum\limits_{n=1}^{\infty}P\left(\left|\sum\limits_{i=1}^{n}X_{ni}\right|＞\epsilon\right)＜ \infty.$

由Borel-Cantelli引理可知, $\forall \epsilon＞0$及当$n$充分大时, 有$\left|\sum\limits_{i=1}^{n}X_{ni}\right|\leq\epsilon$ a.s..从而当$n\rightarrow\infty$时, $\sum\limits_{i=1}^{n}X_{i}\rightarrow 0$ a.s., 即式(3.4) 成立.

最后证式(3.5) 成立.由$Y_{i}^{(2)}$的定义, 根据引理1知$\{(Y_{i}^{(2)}-EY_{i}^{(2)})\}_{i=1}^{n}$仍是零均值同分布ND序列, 故$E\left|Y_{i}^{(2)}-EY_{i}^{(2)}\right|^{2}=\text{Var}(Y_{i}^{(2)})\leq\text{Var}(Y_{i})\leq M$.由Markov不等式得

$\begin{align} &\sum\limits_{i=1}^{\infty }{P}\left( \left| Y_{i}^{(2)}-EY_{i}^{(2)} \right|＞\sqrt{i} \right)=\sum\limits_{i=1}^{\infty }{P}\left( \left| Y_{1}^{(2)}-EY_{1}^{(2)} \right|＞\sqrt{i} \right) \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\sum\limits_{i=1}^{\infty }{\sum\limits_{j=i}^{\infty }{P}}\left( \sqrt{j}＜\left| Y_{1}^{(2)}-EY_{1}^{(2)} \right|＜\sqrt{j+1} \right) \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\sum\limits_{j=1}^{\infty }{\sum\limits_{i=1}^{j}{P}}\left( \sqrt{j}＜\left| Y_{1}^{(2)}-EY_{1}^{(2)} \right|＜\sqrt{j+1} \right) \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\sum\limits_{j=1}^{\infty }{j}P\left( \sqrt{j}＜\left| Y_{1}^{(2)}-EY_{1}^{(2)} \right|＜\sqrt{j+1} \right) \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \le \sum\limits_{j=1}^{\infty }{E}{{\left| Y_{1}^{(2)}-EY_{1}^{(2)} \right|}^{2}}{{I}_{\left( \sqrt{j}＜\left| Y_{1}^{(2)}-EY_{1}^{(2)} \right|＜\sqrt{j+1} \right)}} \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\text{Var}(Y_{1}^{(2)})\le \text{Var}({{Y}_{1}})\le M＜\infty . \\ \end{align}$

(3.7)

由Borel-Cantelli引理可知$\left|Y_{i}^{(2)}-EY_{i}^{(2)}\right|\leq \sqrt{i}$ a.s..记$T_{i}\doteq W_{ni}(t_{0})(Y_{i}^{(2)}-EY_{i}^{(2)})$, 则由$W_{ni}(t_{0})>0$知$\{T_{i}\}_{i=1}^{n}$仍是同分布ND序列, $ET_{i}=0$, 并且

$\mathop {\max }\limits_{1 \le i \le n} \left| {{T_i}} \right| = \mathop {\max }\limits_{1 \le i \le n} {W_{ni}}({t_0})\left| {Y_i^{(2)} - EY_i^{(2)}} \right| \le \frac{{C{\rm{log}}n}}{{n{h_n}}}\sqrt n = \frac{{C{\rm{log}}n}}{{\sqrt n {h_n}}}{\rm{a}}{\rm{.s}}..$

令$t=\frac{\sqrt{n}h_{n}}{C \text{log} n}$, 则有$t\mathop {\max }\limits_{1 \le i \le n} \left| {{T_i}} \right| \le 1$ a.s..从而$\{T_{i}\}_{i=1}^{n}$满足引理3的条件, 再结合条件$\frac{\text{log}^{2} n}{\sqrt{n}h_{n}}\rightarrow 0 (n\rightarrow\infty)$, 故$\forall \epsilon>0$, 有

$\begin{align} &\quad \quad P\left( \left| \sum\limits_{i=1}^{n}{{{T}_{i}}} \right|＞\epsilon \right)\le 2\text{exp}\{-t\epsilon +{{t}^{2}}\sum\limits_{i=1}^{n}{E}T_{i}^{2}\} \\ &=2\text{exp}\left\{ -\frac{\sqrt{n}{{h}_{n}}}{C\text{log}n}\epsilon +{{\left( \frac{\sqrt{n}{{h}_{n}}}{C\text{log}n} \right)}^{2}}\sum\limits_{i=1}^{n}{W_{ni}^{2}}({{t}_{0}})E{{(Y_{i}^{(2)}-EY_{i}^{(2)})}^{2}} \right\} \\ &\le 2\text{exp}\left\{ -\frac{\sqrt{n}{{h}_{n}}}{C\text{log}n}\epsilon +{{\left( \frac{\sqrt{n}{{h}_{n}}}{C\text{log}n} \right)}^{2}}\frac{MC\text{log}n}{n{{h}_{n}}} \right\}=2\text{exp}\left\{ -\frac{\sqrt{n}{{h}_{n}}}{C\text{log}n}\epsilon +\frac{M{{h}_{n}}}{C\text{log}n} \right\} \\ &\le 2\text{exp}\left\{ -\frac{\epsilon }{2}\frac{\sqrt{n}{{h}_{n}}}{C\text{log}n} \right\}\le 2\text{exp}\left\{ \text{log}{{n}^{-2}} \right\}=2{{n}^{-2}}. \\ \end{align}$

(3.8)

故$\forall \epsilon>0$及$n\to \infty$时, 有

$\sum\limits_{n=1}^{\infty}P\left(\left|\sum\limits_{i=1}^{n}T_{i}\right|＞\epsilon\right)＜ \infty, $

此即

$\sum\limits_{n=1}^{\infty}P\left(\left|\sum\limits_{i=1}^{n}W_{ni}(t_{0})(Y_{i}^{(2)}-EY_{i}^{(2)})\right|＞\epsilon\right)＜ \infty.$

由Borel-Cantelli引理得

$\left|\sum\limits_{i=1}^{n}W_{ni}(t_{0})\left(Y_{i}^{(2)}-EY_{i}^{(2)}\right)\right|\leq\epsilon \text{a.s.}, $

即$\sum\limits_{i=1}^{n}W_{ni}(t_{0})\left(Y_{i}^{(2)}-EY_{i}^{(2)}\right)\xrightarrow{\text{a.s.}} 0.$从而式(3.5) 成立, 定理1得证.