Chow^[1]首先提出矩完全收敛概念,它是完全收敛的精细化.许多学者对矩完全收敛性进行了研究,得到了很多深刻的结果,如文献[2-9]等等. Sung在文献[10]中利用$\rho^*$ -混合随机变量序列部分和的最大值的Rosenthal型不等式得到了一个$\rho^*$ -混合随机变量序列加权和(被称为Sung型加权和)的最大值完全收敛性定理,最近Wu等^[2]利用Rosenthal型不等式得到了$\rho^*$ -混合随机变量序列Sung型加权和的最大值矩完全收敛性定理-定理1.1,推广和改进了Sung^[10]的结果.

定理1.1^[2] 设$v>0, \alpha>1/2, \alpha p> 1, q> p\vee v, \{X, X_n\}$是同分布的$\rho^*$ -混合随机变量序列,当$p\vee v\ge 1$时,还设$EX=0$. $\{a_{ni}, 1\le i\le n, n\ge 1\}$是常数阵列且满足

(1.1) $ \begin{equation}\label{eq:1.1}\sum\limits_{i=1}^n |a_{ni}|^q\ll n, \end{equation}$

如果

(1.2) $\left\{ {\begin{array}{*{20}{l}}{E|X{|^p} < \infty , }&{v < p, }\\{E|X{|^p}\log (1 + |X|) < \infty , }&{v = p, }\\{E|X{|^v} < \infty , }&{v > p, }\end{array}} \right. $

则对任意的$\varepsilon>0, $有

(1.3) $ \begin{equation}\label{eq:1.3}\sum\limits_{n=1}^\infty n^{\alpha p-\alpha v-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k a_{ni}X_i\right|-\varepsilon n^{\alpha} \right)_+^v<\infty, \end{equation}$

进而有

(1.4) $ \begin{equation}\label{eq:1.4}\sum\limits_{n=1}^\infty n^{\alpha p-2} P\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k a_{ni}X_i\right|>\varepsilon n^{\alpha} \right)<\infty, \end{equation}$

其中$a\vee b =\max\{a, b\}, x_+=x\vee 0, x_+^v=(x_+)^v$.以下沿用此记号.

事实上,当$0<p\vee v<1$时,从文献[2,定理3.1] (即上面的定理1.1)的证明可以得出(1.3)式与(1.4)式对任意随机变量序列都成立.

本文的目的是:把定理1.1推广到END随机变量序列情形.由于END随机变量序列部分和的最大值的Rosenthal型不等式是否成立目前为止还不清楚,因此本文证明方法与Wu等^[2]有不同,不但在对随机变量的截取上要复杂,而且在定理的证明上(主要体现在引理2.6与引理2.7的证明上)受Li等^[11]的启发,比Wu等^[2]要简便得多.本文所得的结果推广了定理1.1, Chen等^[3], Sung^[10]及Zhang^[12]中的相应的结论.

称随机变量序列$\{X_n, n\ge 1\}$是END的,若存在常数$M\ge 1$,对$\forall~n\ge 1, x_1, x_2, \cdots, x_n\in \mathbb{R} $,都有下二式成立

$ P(X_1>x_1, X_2>x_2, \cdots, X_n>x_n)\le M\prod\limits_{j=1}^n P(X_j>x_j), $

$ P(X_1\le x_1, X_2\le x_2, \cdots, X_n\le x_n)\le M\prod\limits_{j=1}^n P(X_j\le x_j). $

END这一概念是Liu^[13]提出的,以NOD (negatively orthant dependent)^[14]为特殊情形($M=1$). Liu^[13]通过例题指出END是既包含负相依结构又包含正相依结构的非常广泛的概念. Joag-Dev与Proschan^[14]指出NA(negatively associated)列是NOD列,但反之不成立,因此NA列也是END列.邱德华和陈平炎^[8]获得了由END随机变量序列产生的移动平均过程的完全收敛性和矩完全收敛性, Liu^[13]获得了具有重尾的END随机变量的精确大偏差, Qiu等^[15]在不同条件下获得了END阵列加权和的完全收敛性定理, Wang等^[16]研究了END列与END阵列的完全收敛性, Wu和Guan^[17]研究了END列弱大数律及$L_p$收敛性与完全收敛性, Liu^[18]研究了END列中偏差的充要条件, Shen^[19]获得了END列的Rosenthal型不等式并利用它研究了END列的渐近逼近问题,等等.

下面陈述本文主要结果,其证明放在下一节.

定理1.2 设$v>0, \alpha>1/2, p>0, q>p\vee v\ge 1, \alpha( p\vee v)> 1, \{X, X_n\}$是均值为零的同分布的END随机变量序列, $\{a_{ni}, 1\le i\le n, n\ge 1\}$是满足(1.1)式的常数阵列.如果(1.2)式成立,则(1.3)式与(1.4)式成立.

注1.1 1)从本文的证明过程来看,本文的结论对成立最大值的Rosenthal型不等式的随机变量都成立.

2)本文的结论对成立Menshov-Rademacher型不等式的随机变量都成立.令$a_{ni}=1, 1\le i \le n, n\ge 1$,则由定理1.2可得到Chen等^[3]的定理3.7,而且定理1.2还是部分和的最大值情形.

本文以下总用$C$代表与$n$无关的正常数,在不同的地方可表示不同的值,即使在同一式中也是如此, $a\ll b$表示存在正常数$C$使$a\le Cb, $$a\wedge b=\min\{a, b\}$.

为证本文结果,需要如下引理.

引理2.1^[13] 设$\{X_n, n\ge 1\}$是END随机变量序列, $\{f_n, n\ge 1\}$同为单调递增(或同为单调递减)的函数列,则$\{f_n(X_n), n\ge 1\}$仍是END随机变量序列.

引理2.2^[12] 设$\tau\ge 2, \{X_n, n\ge 1\}$是均值为零的END随机变量序列且$E|X_n|^\tau<\infty, $$n\ge 1$.则存在只依赖于$\tau$的正常数$C_\tau$,使

$(ⅰ)\;\;E\left|\sum\limits_{i=1}^{n}X_i\right |^\tau \le C_\tau\left\{\sum\limits_{i=1}^{n} E|X_i|^\tau+ \left(\sum\limits_{i=1}^{n }E|X_i|^2\right)^{\tau/2}\right\};$

$(ⅱ)\;\;E\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^{k}X_i\right |^\tau \le C_\tau\left(\log n\right)^\tau \left\{\sum\limits_{i=1}^{n } E|X_i|^\tau+ \left(\sum\limits_{i=1}^{n }E|X_i|^2\right)^{\tau/2}\right\}.$

完全类似于Chen等^[3]定理2.1与定理2.2的证明可得

引理2.3 设$1< \tau\le 2, \{X_n, n\ge 1\}$是均值为零的END随机变量序列且$E|X_n|^\tau<\infty, $$n\ge 1$.则存在只依赖于$\tau$的正常数$C_\tau$,使

$(ⅰ)\;\;E\left|\sum\limits_{i=1}^{n}X_i\right |^\tau \leq C_\tau\sum\limits_{i=1}^{n } E|X_i|^\tau, $

$(ⅱ)\;\;E\max\limits_{1\leq k\leq n}\left|\sum\limits_{i=1}^{k}X_i\right |^\tau \leq C_\tau\left(\log n\right)^\tau \sum\limits_{i=1}^{n } E|X_i|^\tau.$

引理2.2(ⅱ)与引理2.3(ⅱ)即为所谓的Menshov-Rademacher型矩不等式.

引理2.4^[2] 设$Y$和$Z$是随机变量.则对任意$\tau>v>0, a>0$和$\varepsilon>0$,有

$E\left(|Y+Z|-\varepsilon a\right)_+^v\le C_v\left(\varepsilon^{-\tau}+\frac{v}{\tau}\right)a^{v-\tau}| Y|^\tau+C_v E |Z|^v, $

其中当$0<v\le 1$时$C_v=1$,当$v>1$时$C_v=2^{v-1}.$

根据标准的计算可得

引理2.5 设$\alpha>0, v>0, 0<p<\tau $, $X$是随机变量,则有

$(ⅰ)\;\;\sum\limits_{n = 1}^\infty {{n^{\alpha p - \alpha v - 1}}} E|X{|^v}I(|X| > {n^\alpha }) \ll \left\{ {\begin{array}{*{20}{l}}{E|X{|^p}, }&{v < p, }\\{E|X{|^p}\log (1 + |X|), }&{v = p, }\\{E|X{|^v}, }&{v > p, }\end{array}} \right.$

$(ⅱ)\;\;\;\;\sum\limits_{n = 1}^\infty {{n^{\alpha p - \alpha \tau - 1}}} E|X{|^\tau }I(|X| \le {n^\alpha }) \ll E|X{|^p}.$

引理2.6 设$\alpha>0, v>0, p>0, q>p\vee v, D>0, \{a_{ni}, 1\le i\le n, n\ge 1\}$是常数阵列且满足(1.1)式, $X$是随机变量,则

$\sum\limits_{n = 1}^\infty {{n^{\alpha p - \alpha v - 2}}} \sum\limits_{i = 1}^n E |{a_{ni}}X{|^v}I(|{a_{ni}}X| > D{n^\alpha }) \ll \left\{ {\begin{array}{*{20}{l}}{E|X{|^p}, }&{v < p, }\\{E|X{|^p}\log (1 + |X|), }&{v = p, }\\{E|X{|^v}, }&{v > p.}\end{array}} \right.$

证  由(1.1)式,不失一般性可设

$\sum\limits_{i=1}^n |a_{ni}|^q\le n, \, \, n\ge 1.$

则对任意$\tau:0<\tau\le q, $由上式和Hölder不等式可得

$\begin{equation}\label{eq:2.1} \sum\limits_{k=1}^{n}|a_{nk}|^{\tau}\le \bigg(\sum\limits_{k=1}^{n}|a_{nk}|^{q}\bigg)^{\tau/q} \bigg(\sum\limits_{k=1}^{n}1\bigg)^{1-\tau/q}\le n. \end{equation}$

而对任意$t>0$有

$\begin{eqnarray*} I(|a_{ni}X|>D n^{\alpha})& = &I(|a_{ni}X|>D n^{\alpha}, |X|\le n^\alpha)+I(|a_{ni}X|>D n^{\alpha}, |X|> n^\alpha)\\ & \le& \left(\frac{|a_{ni}X|}{D n^{\alpha}}\right)^{t}I(|X|\le n^\alpha)+I(|X|> n^\alpha), \end{eqnarray*} $

于是再由(2.1)式和$q>p\vee v$及引理2.5可得

$\begin{eqnarray*}& &\sum\limits_{n=1}^\infty n^{\alpha p-\alpha v-2} \sum\limits_{i=1}^n E|a_{ni}X |^v I(|a_{ni}X|>D n^{\alpha} ) \\& \le& D^{v-q}\sum\limits_{n=1}^\infty n^{\alpha p-\alpha q-2} \sum\limits_{i=1}^n E|a_{ni}X |^q I(|X|\le n^\alpha )+\sum\limits_{n=1}^\infty n^{\alpha p-\alpha v-2} \sum\limits_{i=1}^n E|a_{ni}X |^{v} I(|X|> n^\alpha )\\& \le&D^{v-q}\sum\limits_{n=1}^\infty n^{\alpha p-\alpha q-1} E|X |^q I( |X|\le n^\alpha ) + \sum\limits_{n=1}^\infty n^{\alpha p-\alpha v-1} E|X |^{v} I(|X|> n^\alpha )\\& \ll& \left\{\begin{array}{l} E|X|^{p},& v < p, \\ E|X|^{p}\log(1+|X|),& v = p, \\ E|X|^{v},&v > p. \end{array} \right.\end{eqnarray*}$

证毕.

引理2.7 设$\alpha>0, p>0, q>p, D>0$是常数, $\{a_{ni}, 1\le i\le n, n\ge 1\}$是常数阵列且满足(1.1)式, $X$是随机变量,则对任意$\tau>p$有

$\sum\limits_{n=1}^\infty n^{\alpha p-\alpha \tau-2} \sum\limits_{i=1}^n E|a_{ni}X |^\tau I(|a_{ni}X|\le D n^{\alpha} )\ll E|X|^p. $

证  对任意$0<t, \lambda<\tau$有

$\begin{eqnarray*}&& |a_{ni}X |^\tau I(|a_{ni}X|\le D n^{\alpha})\\& = &|a_{ni}X |^\tau \big\{I(|a_{ni}X|\le D n^{\alpha}, |X|\le n^\alpha)+I(|a_{ni}X|\le D n^{\alpha}, |X|> n^\alpha)\big\}\\&\le& \left(D n^{\alpha}\right)^{\tau-t}|a_{ni}X |^t I( |X|\le n^\alpha ) +\left(D n^{\alpha}\right)^{\tau-\lambda}|a_{ni}X |^{\lambda} I(|X|> n^\alpha). \end{eqnarray*}$

取定$t\in (p, \min\{\tau, q\}), \lambda=p/2, $则由上式和(2.1)式及引理2.5可得

$\begin{eqnarray*}&&\sum\limits_{n=1}^\infty n^{\alpha p-\alpha \tau-2} \sum\limits_{i=1}^n E|a_{ni}X |^\tau I(|a_{ni}X|\le D n^{\alpha} )\\& \ll& \sum\limits_{n=1}^\infty n^{\alpha p-\alpha t-2} \sum\limits_{i=1}^n E|a_{ni}X |^t I( |X|\le n^\alpha ) +\sum\limits_{n=1}^\infty n^{-2+\alpha p/2} \sum\limits_{i=1}^n E|a_{ni}X |^{p/2} I(|X|> n^\alpha )\\&\le &\sum\limits_{n=1}^\infty n^{\alpha p-\alpha t-1} E|X |^t I( |X|\le n^\alpha )+\sum\limits_{n=1}^\infty n^{-1+\alpha p/2} E|X |^{p/2} I(|X|> n^\alpha )\\&\ll& E|X|^p. \end{eqnarray*} $

证毕.

定理1.2的证明 先证(1.3)式.对任意给定的$n\ge 1$和任意给定的$\varepsilon>0$,由$C_r$不等式和(1.1)式及(1.2)式可得

$E\left\{\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k a_{ni}X_i \right|-\varepsilon n^{\alpha}\right\}_+^v<\infty.$

因此,要证(1.3)式,只要证明对充分大的正整数$n_0$下式成立

(2.2) $\begin{equation}\label{eq:2.2} \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k a_{ni}X_i\right|-\varepsilon n^{\alpha} \right)_+^v<\infty, ~~\forall \varepsilon>0. \end{equation}$

取定$\theta\in (\frac{1}{\alpha (p\vee v)}, 1)$,对$1\le i\le n, n\ge 1, $令

$X_{ni}^{(1)}=-n^{\alpha\theta} I(a_{ni}X_i<-n^{\alpha\theta} )+a_{ni}X_i I(|a_{ni}X_i|\le n^{\alpha\theta} ) +n^{\alpha\theta} I(a_{ni}X_i> n^{\alpha\theta} ), $

$X_{ni}^{(2)} = ({a_{ni}}{X_i} - {n^{\alpha \theta }})I({n^{\alpha \theta }} < {a_{ni}}{X_i} \le {n^{\alpha \theta }} + {n^\alpha }) + {n^\alpha }I({a_{ni}}{X_i} > {n^{\alpha \theta }} + {n^\alpha }), $

$X_{ni}^{(3)}=(a_{ni}X_i+ n^{\alpha\theta})I(-n^{\alpha\theta}-n^{\alpha}\le a_{ni}X_i<- n^{\alpha\theta} ) - n^{\alpha} I(a_{ni}X_i< -n^{\alpha\theta}-n^{\alpha}), $

$X_{ni}^{(4)}=(a_{ni}X_i- n^{\alpha\theta}-n^{\alpha} )I(a_{ni}X_i> n^{\alpha\theta}+n^{\alpha} ), $

$X_{ni}^{(5)}=(a_{ni}X_i+ n^{\alpha\theta}+n^{\alpha} )I(a_{ni}X_i<- n^{\alpha\theta}-n^{\alpha}). $

则$a_{ni}X_i=\sum\limits_{l=1}^5 X_{ni}^{(l)}.$由$ X_{ni}^{(2)}$的定义, (1.2)式和(2.1)式可得

$\begin{eqnarray*} n^{-\alpha}\max\limits_{1\le k\le n}\left| \sum\limits_{i=1}^k EX_{ni}^{(2)}\right|& =& n^{-\alpha}\sum\limits_{i=1}^n EX_{ni}^{(2)} \\&\le &n^{-\alpha} \sum\limits_{i=1}^nE|a_{ni}X_i| I(|a_{ni}X_i|>n^{\alpha\theta} )\\ & \le& n^{-\alpha} \sum\limits_{i=1}^nE|a_{ni}X_i|\left(\frac{|a_{ni}X_i|}{n^{\alpha \theta}}\right)^{p\vee v-1} I(|a_{ni}X_i|>n^{\alpha\theta} )\\&\ll& n^{1-\alpha\theta(p\vee v)} E|X|^{p\vee v} \to 0, ~~ n\to \infty.\end{eqnarray*} $

由$ X_{ni}^{(4)}$的定义和上式的证明可得

$\begin{eqnarray*} n^{-\alpha}\max\limits_{1\le k\le n}\left| \sum\limits_{i=1}^k EX_{ni}^{(4)}\right|&=&n^{-\alpha} \sum\limits_{i=1}^n EX_{ni}^{(4)} \\&\le &n^{-\alpha} \sum\limits_{i=1}^n E|a_{ni}X_i| I(|a_{ni}X_i|> n^{\alpha} )\to 0, ~~ n\to \infty. \end{eqnarray*}$

同理可证

$\mathop {\lim }\limits_{n \to \infty } n^{-\alpha}\max\limits_{1\le k\le n}\left| \sum\limits_{i=1}^k EX_{ni}^{(3)}\right| =\lim\limits_{n\to \infty} \left(-n^{-\alpha}\sum\limits_{i=1}^n EX_{ni}^{(3)}\right)=0$

和

$\mathop {\lim }\limits_{n \to \infty } n^{-\alpha}\max\limits_{1\le k\le n}\left| \sum\limits_{i=1}^k EX_{ni}^{(5)}\right| =\lim\limits_{n\to \infty} \left(-n^{-\alpha}\sum\limits_{i=1}^n EX_{ni}^{(5)}\right)=0.$

于是,当$n_0$充分大时,由$EX_i=0$,引理2.4和$C_r$不等式可知当$\tau>v$且$v\ge 1$时有

(2.3) $\begin{eqnarray}\label{eq:2.3}&&{\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k a_{ni}X_i\right|-\varepsilon n^{\alpha} \right)_+^v } \nonumber \\& =&\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k \sum\limits_{l=1}^5\left (X_{ni}^{(l)}-EX_{ni}^{(l)}\right)\right|-\varepsilon n^{\alpha} \right)_+^v \nonumber\\&\le&\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left(\sum\limits_{l=1}^5 \max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k \left( X_{ni}^{(l)}-E X_{ni}^{(l)}\right)\right|-\varepsilon n^{\alpha} \right)_+^v \nonumber\\&\le &\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k \left( X_{ni}^{(1)}-E X_{ni}^{(1)}\right)\right|+\sum\limits_{l=2}^5 \left|\sum\limits_{i=1}^n X_{ni}^{(l)}\right| -\varepsilon n^{\alpha}/2 \right)_+^v \nonumber\\&\le& \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k \left( X_{ni}^{(1)}-E X_{ni}^{(1)}\right)\right|+\sum\limits_{l=2}^5 \left|\sum\limits_{i=1}^n \left(X_{ni}^{(l)}-EX_{ni}^{(l)}\right)\right| -\varepsilon n^{\alpha}/3 \right)_+^v \nonumber\\ &\ll& \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k \left( X_{ni}^{(1)}-E X_{ni}^{(1)}\right)\right|^\tau \right) \nonumber\\&\quad& +\sum\limits_{l=2}^3 \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} E\left|\sum\limits_{i=1}^n \left( X_{ni}^{(l)}-E X_{ni}^{(l)}\right)\right|^\tau \nonumber\\&\quad &+\sum\limits_{l=4}^5 \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left|\sum\limits_{i=1}^k \left( X_{ni}^{(l)}-E X_{ni}^{(l)} \right)\right|^v \nonumber\\&=:& I_1+I_2+I_3+I_4+I_5.\end{eqnarray} $

由上证明可知当$\tau>v$且$0<v<1$时有

(2.4) $\begin{eqnarray}\label{eq:2.4}&&{\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k a_{ni}X_i\right|-\varepsilon n^{\alpha} \right)_+^v } \nonumber \\&\le& \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\Bigg(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k \left( X_{ni}^{(1)}-E X_{ni}^{(1)}\right)\right| +\sum\limits_{l=2}^3 \left|\sum\limits_{i=1}^n \left(X_{ni}^{(l)}-EX_{ni}^{(l)}\right)\right| \\&& +\sum\limits_{l=4}^5 \left|\sum\limits_{i=1}^n X_{ni}^{(l)}\right| -\varepsilon n^{\alpha}/3 \Bigg)_+^v \nonumber\\ &\ll& \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} E\left(\max\limits_{1\le k\le n}\left|\sum\limits_{i=1}^k \left( X_{ni}^{(1)}-E X_{ni}^{(1)}\right)\right|^\tau \right) \nonumber\\&& +\sum\limits_{l=2}^3 \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} E\left|\sum\limits_{i=1}^n \left( X_{ni}^{(l)}-E X_{ni}^{(l)}\right)\right|^\tau +\sum\limits_{l=4}^5 \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} E\left|\sum\limits_{i=1}^k X_{ni}^{(l)}\right|^v \nonumber\\&=:& I_1+I_2+I_3+I_4+I_5.\end{eqnarray}$

因此要证(2.2)式,只要证$I_l<\infty, l=1, 2, 3, 4, 5.$再由$C_r$不等式和$a_{ni}=(a_{ni})_+-(a_{ni})_-$知,不失一般性,我们可假设$a_{ni}>0, 1\le i\le n, n\ge 1.$下面分二种情形证明$I_l<\infty, l=1, 2, 3, 4, 5.$

情形一 $1\le p\vee v< 2$.

令$\tau=q\wedge 2$,则由引理2.1,引理2.3, $C_r$不等式, Jensen不等式, $ X_{ni}^{(1)}$的定义, (1.2)式与(2.1)式有

$\begin{eqnarray*} I_1&\ll &\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} (\log n)^\tau \left(\sum\limits_{i=1}^n E |X_{ni}^{(1)}|^\tau \right)\\&\le& \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} (\log n)^\tau \left(\sum\limits_{i=1}^n E |a_{ni}X_{i}|^p n^{(\tau-p)\alpha\theta} \right)\\&\ll& \sum\limits_{n=n_0}^\infty n^{-\alpha (\tau-p)(1-\theta)-1} (\log n)^\tau <\infty.\end{eqnarray*}$

由引理2.1,引理2.3, $ X_{ni}^{(2)}$的定义, $C_r$不等式, Jensen不等式,引理2.6与引理2.7可得

$ \begin{eqnarray*} I_2&\ll&\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} \sum\limits_{i=1}^n E |X_{ni}^{(2)}|^\tau\\&\le &\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} \sum\limits_{i=1}^n \left\{ E |a_{ni}X_i|^\tau I(|a_{ni}X_i|\le 2n^\alpha)+ n^{\alpha\tau} P(|a_{ni}X_i|> n^\alpha)\right\} <\infty.\end{eqnarray*} $

由$ X_{ni}^{(4)}$的定义与引理2.6,当$0<v<1$时再由(2.4)式定义的$I_4$和$C_r$不等式,而当$1\le v<2$时由(2.3)式定义的$I_4$,引理2.1,引理2.3, $C_r$不等式, Jensen不等式可得

(2.5) $ \begin{eqnarray}\label{eq:2.5} I_4&\ll &\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} \sum\limits_{i=1}^n E |X_{ni}^{(4)}|^v \nonumber\\&\le& \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} \sum\limits_{i=1}^n E |a_{ni}X_i|^v I(|a_{ni}X_i|>n^\alpha) <\infty.\end{eqnarray} $

分别类似$I_2<\infty$与$I_4<\infty$可得$I_3<\infty$及$I_5<\infty$,因而(2.2)式在$1\le p\vee v< 2$时成立,故(1.3)式成立.

情形二 $p\vee v\ge 2$.

由(1.2)式有$E|X|^2<\infty$.令$\tau=q\vee \frac{\alpha p}{\alpha-1/2}$,则由引理2.1,引理2.2, $ X_{ni}^{(1)}$的定义, $C_r$不等式, Jensen不等式, (1.2)与(2.1)式有

$\begin{eqnarray*} I_1&\ll &\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} (\log n)^\tau \left\{\sum\limits_{i=1}^n E |X_{ni}^{(1)}|^\tau+\left(\sum\limits_{i=1}^n E (X_{ni}^{(1)})^2 \right)^{\tau/2}\right\}\\&\ll& \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} (\log n)^\tau \left\{\sum\limits_{i=1}^n E |a_{ni}X_{i}|^p n^{(\tau-p)\alpha\theta}+\left(\sum\limits_{i=1}^n E (a_{ni}X_{i})^2 \right)^{\tau/2}\right\}\\&\ll& \sum\limits_{n=n_0}^\infty n^{-\alpha (\tau-p)(1-\theta)-1} (\log n)^\tau+ \sum\limits_{n=n_0}^\infty n^{\alpha p -(\alpha-1/2)\tau-2} (\log n)^\tau<\infty.\end{eqnarray*}$

由引理2.1,引理2.3, $ X_{ni}^{(2)}$的定义, $C_r$不等式, Jensen不等式,引理2.6与引理2.7可得

$ \begin{eqnarray*} I_2&\ll&\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} \left\{\sum\limits_{i=1}^n E |X_{ni}^{(2)}|^\tau+\left(\sum\limits_{i=1}^n E (X_{ni}^{(2)})^2 \right)^{\tau/2}\right\}\\&\ll &\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2} \sum\limits_{i=1}^n \left\{ E |a_{ni}X_i|^\tau I(|a_{ni}X_i|\le 2n^\alpha)+ n^{\alpha\tau} P(|a_{ni}X_i|> n^\alpha)\right\}\\&\quad& +\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha \tau-2}\left(\sum\limits_{i=1}^n E (a_{ni}X_{i})^2 \right)^{\tau/2}\\&\ll& C+\sum\limits_{n=n_0}^\infty n^{\alpha p -(\alpha-1/2)\tau-2} <\infty.\end{eqnarray*} $

对$I_4, $当$0<v\le 2$时,与(2.5)式相同有$I_4<\infty$.当$v>2$时,由引理2.1,引理2.2,引理2.6, $C_r$不等式, Jensen不等式, (1.2)式和(2.1)式可得

$ \begin{eqnarray*} I_4 &\ll &\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} \left\{\sum\limits_{i=1}^n E |X_{ni}^{(4)}|^v+\left(\sum\limits_{i=1}^n E (X_{ni}^{(4)})^2 \right)^{v/2}\right\}\\&\ll&\sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} \left\{ \sum\limits_{i=1}^n E |a_{ni}X_i|^v I(|a_{ni}X_i|>n^\alpha)+\left(\sum\limits_{i=1}^n E (a_{ni}X_i)^2I(|a_{ni}X_i|>n^\alpha) \right)^{v/2}\right\}\\&\ll & C+ \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2} \left(\sum\limits_{i=1}^n E \frac{|a_{ni}X_i|^{p\vee v}}{n^{\alpha((p\vee v)-2)}} I(|a_{ni}X_i|>n^\alpha) \right)^{v/2}\\&\ll &C+ \sum\limits_{n=n_0}^\infty n^{\alpha p-\alpha v-2-\alpha((p\vee v)-2)v/2+v/2} \\&\le &C+ \sum\limits_{n=n_0}^\infty n^{(\alpha (p\vee v)-1)(1- v/2)-1} <\infty.\end{eqnarray*} $

分别类似$I_2<\infty$与$I_4<\infty$的证明可证$I_3<\infty$与$I_5<\infty$,因而(2.2)式在$p\vee v\ge 2$时成立,故(1.3)式成立.而由(1.3)式立得(1.4)式成立.