## On the Complete Moment Convergence for Weighted Sums of Rowwise Asymptotically Almost Negatively Associated Random Variables

Meng Bing1,2, Wang Dingcheng,1, Wu Qunying2

 基金资助: 国家自然科学基金.  11661029国家自然科学基金.  11661030广西自然科学基金.  2018GXNSFAA281011

 Fund supported: the NSFC.  11661029the NSFC.  11661030the Science Foundation of the Guangxi.  2018GXNSFAA281011

Abstract

In this paper, by applying the moment inequality for AANA random sequence, some results on complete moment convergence for weighted sums of AANA random variables are obtained without assumptions of identical distribution, which generalize and improve the corresponding ones of Baek et al. [1] and Wang et al. [12], respectively.

Keywords： AANA random variables ; Complete convergence ; Complete moment convergence ; Weighted sums

Meng Bing, Wang Dingcheng, Wu Qunying. On the Complete Moment Convergence for Weighted Sums of Rowwise Asymptotically Almost Negatively Associated Random Variables. Acta Mathematica Scientia[J], 2020, 40(6): 1670-1681 doi:

## 1 引言

$E|X|^{s}<\infty$, 则对任意的$\varepsilon>0$,

$$$\sum\limits_{n = 1}^\infty n^{\beta}P\bigg(\bigg|\sum\limits_{i = 1}^{\infty}a_{ni}X_{ni}\bigg|>\varepsilon\bigg)<\infty.$$$

$\big({\rm ii}\big) $$1+\alpha+\beta = 0 时, 若 E\big(|X|{\rm log}\big(1+|X|\big)\big)<\infty , 则(1.1)式成立. Wang等[12]把上述结果从行间NA随机变量推广到了AANA随机变量, 而且增加考虑了 1+\alpha+\beta<0 的情况, 获得如下结果. 定理1.2 设 \beta\geq -1 , \left\{X_{ni}; i\geq 1, n\geq 1\right\} 为行间AANA随机变量阵列, 且被随机变量 X 随机控制.假设常数阵列 \left\{a_{ni}; i\geq 1, n\geq 1\right\} 满足如下条件: $$\sup\limits_{i\geq1}\big|a_{ni}\big| = O\big(n^{-r}\big), \; r>0,$$ $$\sum\limits_{i = 1}^{\infty}\big|a_{ni}\big|^{\theta} = O\big(n^{\alpha}\big), \;\mbox{其中}\ 0<\theta<2, \; \theta+\frac{\alpha}{r}<2.$$ \big({\rm i}\big) 假设当 1<\theta<2 时, \sum\limits_{n = 1}^{\infty}\mu^{2}(n)<\infty . 1+\alpha+\beta<0 , E|X|^{\theta}<\infty , 则对任意的 \varepsilon>0 , $$\sum\limits_{n = 1}^\infty n^{\beta}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>\varepsilon\bigg)<\infty,$$ 且(1.1)式也成立. \big({\rm ii}\big)$$ 1+\alpha+\beta = 0$, 且

$$$E|X|^{\theta}{\rm log}|X|<\infty,$$$

$\big({\rm iii}\big) $$1+\alpha+\beta>0 , 设 \beta>-1 , 且当 s = \theta+(1+\alpha+\beta)/r 时, E|X|^{s}<\infty .进一步假设 EX_{ni} = 0 , \sum\limits_{n = 1}^{\infty}\mu^{1/(p-1)}(n)<\infty , p\in \big(3\cdot2^{k-1}, 4\cdot2^{k-1}\big] , 并且 $$p>\max\left\{2, s, \frac{2\big(1+\beta\big)}{r\big(2-\theta\big)-\alpha}\right\},$$ 其中 s\geq1 , 正整数 k\geq1 , 则(1.1)式和(1.4)式成立. 该文在Baek等[1]和Wang等[12]研究结果的基础之上, 将进一步研究不同分布条件下行间AANA随机变量加权和的完全矩收敛.所得结果推广和改进了Baek等[1]和Wang等[12]的结论. 定义1.3 称随机变量序列 \left\{Z_{n}; n\geq 1\right\} 完全矩收敛, 若对 a_{n}>0, b_{n}>0, q>0 , 完全矩收敛的概念是由Chow[3]首先提出的. 定义1.4 称随机变量序列 \left\{X_{n}; n\geq 1\right\} 被随机变量 X 随机控制, 若存在正常数 C , 使得对所有的 n\geq 1 , x\geq 0 , 有 P\big(\big|X_{n}\big|\geq x\big)\leq CP\big(|X|\geq x\big). ## 2 引理 引理2.1[17] 设 \left\{X_{n}; n\geq 1\right\} 是混合系数为 \left\{\mu (n); n\geq 1\right\} 的AANA随机变量序列, 设 \left\{f_{n}; n\geq 1\right\} 是单调不增（或单调不减）的函数序列, 则 \left\{f_{n}\big(X_{n}\big); n\geq 1\right\} 仍是混合系数为 \left\{\mu (n); n\geq 1\right\} 的AANA随机变量序列. 引理2.2[17] 设 \left\{X_{n}; n\geq 1\right\} 是一均值为0的AANA随机变量序列, 当 1<p\leq2 时, 混合系数 \left\{\mu (n); n\geq 1\right\} 满足 \sum\limits_{n = 1}^{\infty}\mu^{2}(n)<\infty , 则存在只依赖于 p 的正常数 C = C_{p} , 使得 $$E\bigg(\max\limits_{1\leq k\leq n}\bigg|\sum\limits_{i = 1}^k X_{i}\bigg|^p \bigg)\leq C\sum\limits_{i = 1}^nE\big|X_{i}\big|^{p}.$$ p\in \big(3\cdot2^{k-1}, 4\cdot2^{k-1}\big] 时, k 是正整数, \sum\limits_{n = 1}^{\infty}\mu^{1/(p-1)}(n)<\infty , 则存在只依赖于 p 的正常数 C = C_{p} , 使得 $$E\bigg(\max\limits_{1\leq k\leq n}\bigg|\sum\limits_{i = 1}^k X_{i}\bigg|^p \bigg)\leq C \left\{\sum\limits_{i = 1}^n E\big|X_{i}\big|^p+\bigg(\sum\limits_{i = 1}^n EX_{i}^2\bigg)^{p/2} \right\}.$$ 引理2.3[16] 设 \{X_{n};n\geq 1\} 为一随机变量序列, 且被随机变量 X 随机控制, 则对所有的 u>0 , t>0 , 下面两式成立: $$E\big|X_{n}\big|^u I\big(\big|X_{n}\big|\leq t\big)\leq C\left[E|X|^u I\big(|X|\leq t\big)+t^u P\big(|X|>t\big)\right],$$ $$E\big|X_{n}\big|^u I\big(\big|X_{n}\big|> t\big)\leq CE|X|^u I\big(|X|> t\big).$$ ## 3 主要结果及证明 整篇文章中, \left\{X_{ni};i\geq1, n\geq1\right\} 为行间AANA阵列, 且混合系数为 \left\{\mu(i), i\geq1\right\} .正常数 C 在不同的地方有所不同. 定理3.1 设 \beta\geq -1 , \left\{X_{ni};i\geq1, n\geq1\right\} 为行间AANA阵列, 且被随机变量 X 随机控制. \left\{a_{ni};i\geq1, n\geq1\right\} 为满足(1.2)式, (1.3)式的常数阵列. \big({\rm i}\big)$$ 1<\theta<2$, $\sum\limits_{n = 1}^{\infty}\mu^{2}(n)<\infty$.$1+\alpha+\beta<0$, $E|X|^{\theta}<\infty$, 则对任意的$\varepsilon>0$,

$$$\sum\limits_{n = 1}^\infty n^{\beta}E\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|-\varepsilon\bigg)_{+}^{\theta}<\infty,$$$

$$$\sum\limits_{n = 1}^\infty n^{\beta}E\bigg(\bigg|\sum\limits_{i = 1}^{\infty}a_{ni}X_{ni}\bigg|-\varepsilon\bigg)_{+}^{\theta}<\infty.$$$

$\big({\rm ii}\big) $$1+\alpha+\beta = 0 , 且 $$E|X|^{\theta}{\rm log}|X|<\infty,$$ 进一步假设 EX_{ni} = 0 , 当 1\leq\theta<2 时, \sum\limits_{n = 1}^{\infty}\mu^{2}(n)<\infty , 则(3.1)式和(3.2)式成立. \big({\rm iii}\big)$$ 1+\alpha+\beta>0$, $\beta>-1$以及

$$$E|X|^{s}<\infty, \; s = \theta+\frac{1+\alpha+\beta}{r}.$$$

$$$p>\max\left\{2, s, \frac{2\big(1+\beta\big)}{r\big(2-\theta\big)-\alpha}\right\},$$$

$\begin{eqnarray} \infty &>&\sum\limits_{n = 1}^\infty n^{\beta}E\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|-\varepsilon\bigg)_{+}^{\theta}\\ & = &\sum\limits_{n = 1}^\infty n^{\beta}\int_{0}^{\infty}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|-\varepsilon>x^{1/ \theta}\bigg){\rm d}x\\ &\geq&C\int_{0}^{1}\sum\limits_{n = 1}^\infty n^{\beta}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>\varepsilon\bigg){\rm d}x\\ & = &C\sum\limits_{n = 1}^\infty n^{\beta}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>\varepsilon\bigg). \end{eqnarray}$

(3.2)式的证明完全类似于(3.1)式的证明, 因此, 该文仅证明(3.1)式成立.不失一般性, 设$a_{ni}>0$, $n, i\geq1$ (否则, 用$a_{ni}^{+} $$a_{ni}^{-} 替代 a_{ni} , 则 a_{ni} = a_{ni}^{+}-a_{ni}^{-} ).由(1.2)式和(1.3)式, 不妨设 { } \sup_{i\geq1}\big|a_{ni}\big|\leq n^{-r}$$ \sum\limits_{i = 1}^{\infty}\big|a_{ni}\big|^{\theta}\leq n^{\alpha}$.对任意$\varepsilon>0$,

$\begin{eqnarray} && \sum\limits_{n = 1}^\infty n^{\beta}E\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|-\varepsilon\bigg)_{+}^{\theta} = \sum\limits_{n = 1}^{\infty} n^{\beta}\int_{0}^{1}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>\varepsilon+x^{1/ \theta}\bigg){\rm d}x \\ &&+\sum\limits_{n = 1}^{\infty} n^{\beta}\int_{1}^{\infty}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>\varepsilon+x^{1/ \theta}\bigg){\rm d}x \\ &\leq&\sum\limits_{n = 1}^\infty n^{\beta}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>\varepsilon\bigg) +\sum\limits_{n = 1}^{\infty} n^{\beta}\int_{1}^{\infty}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>x^{1/ \theta}\bigg){\rm d}x \\ &\triangleq& I+J. \end{eqnarray}$

$\big({\rm i}\big) $$1+\alpha+\beta<0 注意到: $$\int_{a}^{\infty}P\big(\big|Y\big|>x^{1/ \theta}\big){\rm d}x\leq E\big|Y\big|^{\theta}I\big(\big|Y\big| >a^{1/ \theta}\big).$$ 由(3.8)式, (2.1)式和(1.3)式可知 \begin{eqnarray} J &\leq& \sum\limits_{n = 1}^\infty n^{\beta}E\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|^{\theta}\bigg) I\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|^{\theta}>1\bigg)\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}E\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|^{\theta}\bigg)\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n\big|a_{ni}\big|^{\theta}E|X|^{\theta} \leq C\sum\limits_{n = 1}^\infty n^{\alpha+\beta}E|X|^{\theta} \leq CE|X|^{\theta}<\infty. \end{eqnarray} 接下来证明当 1+\alpha+\beta\geq0 时, J<\infty .对固定的 n\geq1 , 定义 由引理2.1可知, \left\{Y_{ni}; i\geq1, n\geq 1\right\} 仍是AANA随机阵列, 容易验证 \begin{eqnarray} J & = &\sum\limits_{n = 1}^{\infty} n^{\beta}\int_{1}^{\infty}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}a_{ni}X_{ni}\bigg|>x^{1/ \theta}\bigg){\rm d}x\\ &\leq& \sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n\int_{1}^{\infty}P\big(\big|a_{ni}X_{ni}\big|>x^{1/ \theta}\big){\rm d}x +\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}Y_{ni}\bigg|>x^{1/ \theta}\bigg){\rm d}x\\ &\triangleq& J_{1}+J_{2}. \end{eqnarray} 因此, 要证(3.1)式成立, 只需证明 J_{1}<\infty , J_{2}<\infty . \big({\rm ii}\big)$$ 1+\alpha+\beta = 0$

$\begin{eqnarray} J_{1} &\leq& \sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n E\big|a_{ni}X_{ni}\big|^{\theta}I\big(\big|a_{ni}X_{ni}\big|>1\big)\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n\big|a_{ni}\big|^{\theta}E|X|^{\theta} I\big(|X|>\big|a_{ni}\big|^{-1}\big)\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\alpha+\beta}E|X|^{\theta}I\big(|X|>n^{r}\big)\\ & = & C\sum\limits_{n = 1}^\infty n^{-1}\sum\limits_{k = n}^\infty E|X|^{\theta} I\big(k^{r}<|X|\leq\big(k+1\big)^{r}\big)\\ &\leq& C\sum\limits_{k = 1}^\infty E|X|^{\theta}I\big(k^{r}<|X|\leq\big(k+1\big)^{r}\big) \sum\limits_{n = 1}^kn^{-1}\\ &\leq& C\sum\limits_{k = 1}^\infty {\rm log}{k}E|X|^{\theta}I\big(k^{r}<|X|\leq\big(k+1\big)^{r}\big)\\ &\leq& CE|X|^{\theta}{\rm log}{|X|}<\infty. \end{eqnarray}$

$\begin{eqnarray} J_{1} & = &\sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n\int_{1}^{\infty}P\big(\big|a_{ni}X_{ni}\big|>x^{1/ \theta}\big){\rm d}x\\ &\leq& \sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^nE\big|a_{ni}X\big|^{\theta}I\big(\big|a_{ni}X\big|>1\big)\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\alpha+\beta}E|X|^{\theta}I\big(|X|>n^{r}\big)\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\alpha+\beta}\sum\limits_{k = n}^\infty E|X|^{\theta}I\big(k^{r}<|X|\leq \big(k+1\big)^{r}\big)\\ &\leq& C\sum\limits_{k = 1}^\infty E|X|^{\theta}I\big(k^{r}<|X|\leq \big(k+1\big)^{r}\big)\sum\limits_{n = 1}^k n^{\alpha+\beta}\\ &\leq& C\sum\limits_{k = 1}^\infty k^{1+\alpha+\beta}E|X|^{\theta}I\big(k^{r}<|X|\leq \big(k+1\big)^{r}\big)\\ &\leq& CE|X|^{\theta+(1+\alpha+\beta)/r}<\infty. \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{x\geq 1}x^{-1/\theta}\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}EY_{ni}\bigg| &\leq& C\sup\limits_{x\geq 1}x^{-1/\theta}\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}Ea_{ni}X_{ni}I\big(\big|a_{ni}X_{ni}\big|>x^{1/\theta}\big)\bigg|\\ &&+ C\sup\limits_{x\geq 1}x^{-1/\theta}\sum\limits_{i = 1}^{n}x^{1/\theta}P\big(\big|a_{ni}X_{ni}\big|>x^{1/\theta}\big)\\ &\leq& C\sup\limits_{x\geq 1}x^{-1/\theta}\sum\limits_{i = 1}^{n}E\big|a_{ni}X_{ni}\big|I\big(\big|a_{ni}X_{ni}\big|>x^{1/\theta}\big)\\ &\leq& C\sup\limits_{x\geq 1}x^{-s/\theta}\sum\limits_{i = 1}^{n}E\big|a_{ni}X\big|^{s}I\big(\big|a_{ni}X\big|>x^{1/\theta}\big)\\ &\leq& C\sup\limits_{i\geq 1}\big|a_{ni}\big|^{s-\theta}\sum\limits_{i = 1}^{n}\big|a_{ni}\big|^{\theta}E|X|^{s} I\big(\big|a_{ni}X\big|>1\big)\\ &\leq& Cn^{\alpha-\big(s-\theta\big)r}E|X|^{s}I\big(|X|>n^{r}\big)\\ &\leq& Cn^{-1-\beta}E|X|^{s}I\big(|X|>n^{r}\big)\rightarrow 0, \; n\rightarrow\infty, \end{eqnarray}$

$$$p>\max\left\{2, \theta+\frac{1+\alpha+\beta}{r}, \frac{2\big(1+\beta\big)}{r\big(2-\theta\big)-\alpha}\right\}.$$$

$\begin{eqnarray} J_{2} & = &\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}Y_{ni}\bigg| >x^{1/ \theta}\bigg){\rm d}x\\ &\leq& \sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}P\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}\big(Y_{ni}-EY_{ni}\big)\bigg|>\frac {x^{1/ \theta}}{2}\bigg){\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}x^{-p/\theta}E\bigg(\max\limits_{1\leq j\leq n}\bigg|\sum\limits_{i = 1}^{j}\big(Y_{ni}-EY_{ni}\big)\bigg|^{p}\bigg){\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n\int_{1}^{\infty}x^{-p/\theta}E\big|Y_{ni}-EY_{ni}\big|^p{\rm d}x\\ &&+C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}x^{-p/\theta}\bigg(\sum\limits_{i = 1}^nE\big(Y_{ni}-EY_{ni}\big)^2\bigg)^{p/2}{\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n\int_{1}^{\infty}x^{-p/\theta}E\big|Y_{ni}\big|^p{\rm d}x +C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}x^{-p/\theta}\bigg(\sum\limits_{i = 1}^nEY_{ni}^2\bigg)^{p/2}{\rm d}x\\ &\triangleq& J_{2}^{(1)}+J_{2}^{(2)}. \end{eqnarray}$

$\begin{eqnarray} J_{2}^{(2)} & = &C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}x^{-p/\theta}\bigg(\sum\limits_{i = 1}^nEY_{ni}^2\bigg)^{p/2}{\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}x^{-p/\theta}\bigg(\sum\limits_{i = 1}^nE\big|a_{ni}X_{ni}\big|^{2} I\big(\big|a_{ni}X_{ni}\big|\leq x^{1/\theta}\big) \bigg)^{p/2}{\rm d}x\\ &&+C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}\bigg(\sum\limits_{i = 1}^nP\big(\big|a_{ni}X_{ni}\big|>x^{1/\theta}\big)\bigg) ^{p/2}{\rm d}x\\ &\triangleq& J_{21}^{(2)}+J_{22}^{(2)}. \end{eqnarray}$

$\begin{eqnarray} J_{21}^{(2)} &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\bigg(\int_{1}^{\infty}x^{-p/\theta}{\rm d}x\bigg)\bigg(\sum\limits_{i = 1}^n\big|a_{ni}\big|^2 E|X|^{2}\bigg)^{p/2}\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\bigg(\sum\limits_{i = 1}^n\big|a_{ni}\big|^{\theta} \sup\limits_{i\geq1}\big|a_{ni}\big|^{2-\theta}\bigg)^{p/2}\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta+\left[\alpha-r(2-\theta)\right]p/2}<\infty. \end{eqnarray}$

$1\leq s<2$时, 由$p>2$, $E|X|^s<\infty$可得

$\begin{eqnarray} J_{21}^{(2)} &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}\bigg(\sum\limits_{i = 1}^nE\big|x^{-1/\theta}a_{ni}X_{ni}\big|^2 I\big(\big|a_{ni}X_{ni}\big|\leq x^{1/\theta}\big)\bigg)^{p/2}{\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}\bigg(\sum\limits_{i = 1}^nE\big|x^{-1/\theta}a_{ni}X_{ni}\big|^s I\big(\big|a_{ni}X_{ni}\big|\leq x^{1/\theta}\big)\bigg)^{p/2}{\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\int_{1}^{\infty}x^{-sp/2\theta}\bigg(\sum\limits_{i = 1}^nE\big|a_{ni}X_{ni}\big|^s I\big(\big|a_{ni}X_{ni}\big|\leq x^{1/\theta}\big)\bigg)^{p/2}{\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\bigg(\int_{1}^{\infty}x^{-sp/2\theta}{\rm d}x\bigg)\bigg(\sum\limits_{i = 1}^n\big|a_{ni}\big|^s E|X|^s\bigg)^{p/2}\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\bigg(\sum\limits_{i = 1}^n\big|a_{ni}\big|^\theta\sup\limits_{i\geq1}\big|a_{ni}\big|^{s-\theta}\bigg) ^{p/2}\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta+(r\theta+\alpha-rs)p/2} = C\sum\limits_{n = 1}^\infty n^{\beta-(1+\beta)p/2}<\infty. \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{x\geq1}\bigg|\sum\limits_{i = 1}^nP\big(\big|a_{ni}X_{ni}\big|>x^{1/\theta}\big)\bigg| &\leq& C\sum\limits_{i = 1}^nP\big(\big|a_{ni}X\big|>1\big)\\ &\leq& C\sum\limits_{i = 1}^nE\big|a_{ni}X\big|^sI\big(\big|a_{ni}X\big|>1\big)\\ &\leq& C\sup\limits_{i\geq1}\big|a_{ni}\big|^{s-\theta}\sum\limits_{i = 1}^n\big|a_{ni}\big|^{\theta}E|X|^s I\big(\big|a_{ni}X\big|>1\big)\\ &\leq& Cn^{\alpha-(s-\theta)r}E|X|^sI\big(|X|>n^{r}\big)\\ &\leq& Cn^{-1-\beta}E|X|^sI\big(|X|>n^{r}\big)\rightarrow 0, \; n\rightarrow\infty. \end{eqnarray}$

$\begin{eqnarray} J_{22}^{(2)} &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^n\int_{1}^{\infty}P\big(\big|a_{ni}X_{ni}\big|>x^{1/\theta}\big){\rm d}x\\ &\leq& C\sum\limits_{n = 1}^\infty n^{\beta}\sum\limits_{i = 1}^nE\big|a_{ni}X\big|^{\theta}I\big(\big|a_{ni}X\big|>1\big)\\ &\leq& CE|X|^{\theta+(1+\alpha+\beta)/r}<\infty. \end{eqnarray}$

$p>0$, 使得$\theta+(1+\alpha+\beta)/r = s<p<1$, 则$\alpha+\beta-r(p-\theta)<-1$.类似$1+\alpha+\beta = 0$中情况1的证明, 容易得到: $J_{21}^{*}\leq CE|X|^{\theta+(1+\alpha+\beta)/r}<\infty$, $J_{21}^{**}\leq CE|X|^{\theta+(1+\alpha+\beta)/r}<\infty$, $J_{22}\leq CE|X|^{\theta+(1+\alpha+\beta)/r}<\infty$以及$J_{23}\leq CE|X|^{\theta+(1+\alpha+\beta)/r}<\infty$.证毕.

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