# Mean convergence theorems for weighted sums of random variables under a condition of weighted integrability

## Abstract

In this paper, a new concept of weighted integrability is introduced for an array of random variables concerning an array of constants, which is weaker than other previous related notions of integrability. Mean convergence theorems for weighted sums of an array of dependent random variables satisfying this condition of integrability are obtained. Our results extend and sharpen the known results in the literature.

MSC:60F15.

## 1 Introduction

The notion of uniform integrability plays the central role in establishing weak laws of large numbers. In this paper, we introduce a new notion of weighted integrability and prove some weak laws of large numbers under this condition.

Definition 1.1 A sequence $\left\{{X}_{n},nâ‰¥1\right\}$ of integrable random variables is said to be uniformly integrable if

$\underset{aâ†’\mathrm{âˆž}}{lim}\underset{nâ‰¥1}{sup}E|{X}_{n}|I\left(|{X}_{n}|>a\right)=0.$

Landers and Rogge [1] proved the weak law of large numbers under the sequence of pairwise independent uniformly integrable random variables.

Chandra [2] obtained the weak law of large numbers under a new condition which is weaker than uniform integrability: CesÃ ro uniform integrability.

Definition 1.2 A sequence $\left\{{X}_{n},nâ‰¥1\right\}$ of integrable random variables is said to be CesÃ ro uniformly integrable if

$\underset{aâ†’\mathrm{âˆž}}{lim}\underset{nâ‰¥1}{sup}\frac{1}{n}\underset{i=1}{\overset{n}{âˆ‘}}E|{X}_{i}|I\left(|{X}_{i}|>a\right)=0.$

In the following, let $\left\{{u}_{n},nâ‰¥1\right\}$ and $\left\{{v}_{n},nâ‰¥1\right\}$ be two sequences of integers (not necessary positive or finite) such that ${v}_{n}>{u}_{n}$ for all $nâ‰¥1$ and ${v}_{n}âˆ’{u}_{n}â†’\mathrm{âˆž}$ as $nâ†’\mathrm{âˆž}$. Let $\left\{{k}_{n},nâ‰¥1\right\}$ be a sequence of positive numbers such that ${k}_{n}â†’\mathrm{âˆž}$ as $nâ†’\mathrm{âˆž}$.

OrdÃ³Ã±ez Cabrera [3] introduced the concept of uniform integrability concerning an array of constant weights.

Definition 1.3 Let $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of random variables and $\left\{{a}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of constants with ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}|{a}_{ni}|â‰¤C$ for all $nâˆˆN$ and some constant $C>0$. The array $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is said to be $\left\{{a}_{ni}\right\}$-uniformly integrable if

$\underset{aâ†’\mathrm{âˆž}}{lim}\underset{nâ‰¥1}{sup}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}|E|{X}_{ni}|I\left(|{X}_{ni}|>a\right)=0.$

OrdÃ³Ã±ez Cabrera [3] proved that the condition of uniform integrability concerning an array of constant weights is weaker than uniform integrability, and leads to CesÃ ro uniform integrability as a special case. Under the condition of uniform integrability concerning the weights, he obtained the weak law of large numbers for weighted sums of pairwise independent random variables.

Sung [4] introduced the concept of CesÃ ro-type uniform integrability with exponent r.

Definition 1.4 Let $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of random variables and $r>0$. The array $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is said to be CesÃ ro-type uniformly integrable with exponent r if

$\underset{nâ‰¥1}{sup}\frac{1}{{k}_{n}}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}E{|{X}_{ni}|}^{r}<\mathrm{âˆž}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{aâ†’\mathrm{âˆž}}{lim}\underset{nâ‰¥1}{sup}\frac{1}{{k}_{n}}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}E{|{X}_{ni}|}^{r}I\left({|{X}_{ni}|}^{r}>a\right)=0.$

Note that the conditions of CesÃ ro uniform integrability and CesÃ ro-type uniform integrability with exponent r are equivalent when ${u}_{n}=1$, ${v}_{n}=n$, $nâ‰¥1$ and $r=1$. Sung [4] obtained the weak law of large numbers for an array $\left\{{X}_{ni}\right\}$ satisfying CesÃ ro-type uniform integrability with exponent r for some $0.

Chandra and Goswami [5] introduced the concept of CesÃ ro Î±-integrability ($\mathrm{Î±}>0$) and showed that CesÃ ro Î±-integrability, for any $\mathrm{Î±}>0$, is weaker than CesÃ ro uniform integrability.

Definition 1.5 Let $\mathrm{Î±}>0$. A sequence $\left\{{X}_{n},nâ‰¥1\right\}$ of random variables is said to be CesÃ ro Î±-integrable if

$\underset{nâ‰¥1}{sup}\frac{1}{n}\underset{i=1}{\overset{n}{âˆ‘}}E|{X}_{i}|<\mathrm{âˆž}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{n}\underset{i=1}{\overset{n}{âˆ‘}}E|{X}_{i}|I\left(|{X}_{i}|>{i}^{\mathrm{Î±}}\right)=0.$

Under the CesÃ ro Î±-integrability condition for some $\mathrm{Î±}>\frac{1}{2}$, Chandra and Goswami [5] obtained the weak law of large numbers for a sequence of pairwise independent random variables. They also proved that CesÃ ro Î±-integrability for appropriate Î± is also sufficient for the weak law of large numbers to hold for certain special dependent sequences of random variables.

OrdÃ³Ã±ez Cabrera and Volodin [6] introduced the notion of h-integrability for an array of random variables concerning an array of constant weights, and proved that this concept is weaker than CesÃ ro uniform integrability, $\left\{{a}_{ni}\right\}$-uniform integrability and CesÃ ro Î±-integrability.

Definition 1.6 Let $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of random variables and $\left\{{a}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of constants with ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}|{a}_{ni}|â‰¤C$ for all $nâˆˆN$ and some constant $C>0$. Moreover, let $\left\{h\left(n\right),nâ‰¥1\right\}$ be an increasing sequence of positive constants with $h\left(n\right)â†‘\mathrm{âˆž}$ as $nâ†‘\mathrm{âˆž}$. The array $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is said to be h-integrable with respect to the array of constants $\left\{{a}_{ni}\right\}$ if

$\underset{nâ‰¥1}{sup}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}|E|{X}_{ni}|<\mathrm{âˆž}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{nâ†’\mathrm{âˆž}}{lim}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}|E|{X}_{ni}|I\left(|{X}_{ni}|>h\left(n\right)\right)=0.$

Under appropriate conditions on the weights, OrdÃ³Ã±ez Cabrera and Volodin [6] proved that h-integrability concerning the weights is sufficient for the weak law of large numbers to hold for weighted sums of an array of random variables, when these random variables are subject to some special kind of rowwise dependence.

Sung et al. [7] introduced the notion of h-integrability with exponent r ($r>0$).

Definition 1.7 Let $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of random variables and $r>0$. Moreover, let $\left\{h\left(n\right),nâ‰¥1\right\}$ be an increasing sequence of positive constants with $h\left(n\right)â†‘\mathrm{âˆž}$ as $nâ†‘\mathrm{âˆž}$. The array $\left\{{X}_{ni}\right\}$ is said to be h-integrable with exponent r if

$\underset{nâ‰¥1}{sup}\frac{1}{{k}_{n}}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}E{|{X}_{ni}|}^{r}<\mathrm{âˆž}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{{k}_{n}}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}E{|{X}_{ni}|}^{r}I\left({|{X}_{ni}|}^{r}>h\left(n\right)\right)=0.$

Sung et al. [7] proved that h-integrability with exponent r ($r>0$) is weaker than CesÃ ro-type uniform integrability with exponent r, and obtained weak law of large numbers for an array of dependent random variables (martingale difference sequence or negatively associated random variables ) satisfying the condition of h-integrability with exponent r.

Chandra and Goswami [8] introduced the concept of residual CesÃ ro $\left(\mathrm{Î±},p\right)$- integrability ($\mathrm{Î±}>0$, $p>0$) and showed that residual CesÃ ro $\left(\mathrm{Î±},p\right)$-integrability for any $\mathrm{Î±}>0$ is strictly weaker than CesÃ ro Î±-integrability.

Definition 1.8 Let $\mathrm{Î±}>0$, $p>0$. A sequence $\left\{{X}_{n},nâ‰¥1\right\}$ of random variables is said to be residually CesÃ ro $\left(\mathrm{Î±},p\right)$-integrable if

$\underset{nâ‰¥1}{sup}\frac{1}{n}\underset{i=1}{\overset{n}{âˆ‘}}E{|{X}_{i}|}^{p}<\mathrm{âˆž}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{n}\underset{i=1}{\overset{n}{âˆ‘}}E{\left(|{X}_{i}|âˆ’{i}^{\mathrm{Î±}}\right)}^{p}I\left(|{X}_{i}|>{i}^{\mathrm{Î±}}\right)=0.$

Under the residual CesÃ ro $\left(\mathrm{Î±},p\right)$-integrability condition for some appropriate Î± and p, Chandra and Goswami [8] obtained ${L}^{1}$-convergence and the weak law of large numbers for a sequence of dependent random variables.

We now introduce a new concept of integrability.

Definition 1.9 Let $r>0$ and $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of random variables. Moreover, let $\left\{{a}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of constants and $\left\{h\left(n\right),nâ‰¥1\right\}$ an increasing sequence of positive constants with $h\left(n\right)â†‘\mathrm{âˆž}$ as $nâ†‘\mathrm{âˆž}$. The array $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is said to be residually $\left(r,h\right)$-integrable with respect to the array of constants $\left\{{a}_{ni}\right\}$ if

$\underset{nâ‰¥1}{sup}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}|}^{r}<\mathrm{âˆž}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{nâ†’\mathrm{âˆž}}{lim}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{\left(|{X}_{ni}|âˆ’h\left(n\right)\right)}^{r}I\left(|{X}_{ni}|>h\left(n\right)\right)=0.$

Remark 1.1

1. (i)

The residual $\left(1,h\right)$-integrability concerning the arrays of constants was defined by Yuan and Tao [9], who called it the residual h-integrability, and was extended by OrdÃ³Ã±ez Cabrera et al. [10] to the conditionally residually h-integrability relative to a sequence of Ïƒ-algebras.

2. (ii)

If $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is h-integrable with exponent r, then it is residually $\left(r,{h}^{1/r}\right)$-integrable with respect to the array of constants $\left\{{a}_{ni}\right\}$ satisfying ${a}_{ni}={k}_{n}^{âˆ’1/r}$, ${u}_{n}â‰¤iâ‰¤{v}_{n}$, $nâ‰¥1$.

3. (iii)

Residually $\left(r,h\right)$-integrable with respect to the array of constants $\left\{{a}_{ni}\right\}$ is weaker than residually CesÃ ro $\left(\mathrm{Î±},p\right)$-integrable.

4. (iv)

The concept of residually $\left(r,h\right)$-integrable concerning the array of constants $\left\{{a}_{ni}\right\}$ is strictly weaker than the concept of h-integrable concerning the array of constants $\left\{{a}_{ni}\right\}$ and h-integrable with exponent r.

Therefore, the concept of residually $\left(r,h\right)$-integrable concerning the array of constants $\left\{{a}_{ni}\right\}$ is weaker than the concept of all Definitions 1.1-1.7, and leads to residual CesÃ ro $\left(\mathrm{Î±},p\right)$-integrability as a special case.

For the array $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ of random variables, weak laws of large numbers have been established by many authors (referring to: Sung et al. [7, 11]; Sung [4]; OrdÃ³Ã±ez Cabrera and Volodin [6]).

In this paper, we obtain weak laws of large numbers for the array of dependent random variables satisfying the condition of residually $\left(r,h\right)$-integrable with respect to the array of constants $\left\{{a}_{ni}\right\}$. Our results extend and sharpen the results of Sung et al. [7], Sung et al. [11], Sung [4], OrdÃ³Ã±ez Cabrera and Volodin [6].

## 2 Preliminary lemmas

In order to consider the mean convergence for an array of random variables satisfying dependent conditions, we need the following definition.

Definition 2.1 Two random variables X and Y are said to be negatively quadrant dependent (NQD) or lower case negatively dependent (LCND) if

$P\left(Xâ‰¤x,Yâ‰¤y\right)â‰¤P\left(Xâ‰¤x\right)P\left(Yâ‰¤y\right)\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆR.$

An infinite family of random variables $\left\{{X}_{n},nâ‰¥1\right\}$ is said to be pairwise NQD if every two random variables ${X}_{i}$ and ${X}_{j}$ () are NQD. The array $\left\{{X}_{ni},iâ‰¥1,nâ‰¥1\right\}$ is said to be rowwise pairwise NQD if every positive integer n, the sequence of random variables $\left\{{X}_{ni},iâ‰¥1\right\}$ is pairwise NQD.

This definition was introduced by Alam and Saxena [12] and carefully studied by Joag-Dev and Proschan [13].

Lemma 2.1 Let $\left\{{X}_{n},nâ‰¥1\right\}$ be a sequence of pairwise NQD random variables. Let $\left\{{f}_{n},nâ‰¥1\right\}$ be a sequence of increasing functions. Then $\left\{{f}_{n}\left({X}_{n}\right),nâ‰¥1\right\}$ is a sequence of pairwise NQD random variables.

If random variables X and Y are NQD, then $E\left(XY\right)â‰¤EXEY$, so we have the following.

Lemma 2.2 Let $\left\{{X}_{n},nâ‰¥1\right\}$ be a sequence of pairwise NQD random variables with $E{X}_{n}=0$ and $E{X}_{n}^{2}<\mathrm{âˆž}$, $nâ‰¥1$. Then

$E{\left(\underset{i=1}{\overset{n}{âˆ‘}}{X}_{i}\right)}^{2}â‰¤\underset{i=1}{\overset{n}{âˆ‘}}E{X}_{i}^{2}.$

Using the above lemma, Chen et al. [14] obtained the following inequality.

Lemma 2.3 Let $\left\{{X}_{n},nâ‰¥1\right\}$ be a sequence of pairwise NQD random variables with $E{X}_{n}=0$ and $E{|{X}_{n}|}^{p}<\mathrm{âˆž}$, $nâ‰¥1$, where $1â‰¤pâ‰¤2$. Then

$E{|\underset{i=1}{\overset{n}{âˆ‘}}{X}_{i}|}^{p}â‰¤{c}_{p}\underset{i=1}{\overset{n}{âˆ‘}}E{|{X}_{i}|}^{p},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâ‰¥2,$

where ${c}_{p}>0$ depends only on p.

## 3 Main results and proofs

Theorem 3.1 Let $0 and $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of random variables. Let $\left\{{a}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of constants and $\left\{h\left(n\right),nâ‰¥1\right\}$ an increasing sequence of positive constants with $h\left(n\right)â†‘\mathrm{âˆž}$ as $nâ†‘\mathrm{âˆž}$. Assume that the following conditions hold:

1. (i)

$\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is residually $\left(r,h\right)$-integrable concerning the array $\left\{{a}_{ni}\right\}$;

2. (ii)

$h\left(n\right){sup}_{{u}_{n}â‰¤iâ‰¤{v}_{n}}|{a}_{ni}|â†’0$.

Then

$\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left({X}_{ni}âˆ’{b}_{ni}\right)â†’0$

in ${L}^{r}$ and, hence, in probability as $nâ†’\mathrm{âˆž}$, where ${b}_{ni}=0$ if $0 and ${b}_{ni}=E\left({X}_{ni}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)$ if $1â‰¤r<2$, where ${\mathrm{â„‘}}_{n,i}=\mathrm{Ïƒ}\left({X}_{ni},{u}_{n}â‰¤jâ‰¤i\right)$, ${u}_{n}â‰¤iâ‰¤{v}_{n}$, $nâ‰¥1$, and ${\mathrm{â„‘}}_{n,{u}_{n}âˆ’1}=\left\{\mathrm{âˆ\dots },\mathrm{Î©}\right\}$, $nâ‰¥1$.

Proof If ${u}_{n}=âˆ’\mathrm{âˆž}$ and/or ${v}_{n}=+\mathrm{âˆž}$, by the ${C}_{r}$-inequality, Jensenâ€™s inequality and $0, we have

$\underset{nâ‰¥1}{sup}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}âˆ’{b}_{ni}|}^{r}â‰¤2\underset{nâ‰¥1}{sup}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}|}^{r}<\mathrm{âˆž}.$

Therefore, if $0, we have $E{\left({âˆ‘}_{i={u}_{n}}^{{v}_{n}}|{a}_{ni}{X}_{ni}|\right)}^{r}â‰¤{âˆ‘}_{i={u}_{n}}^{{v}_{n}}{|{a}_{ni}|}^{r}E{|{X}_{ni}|}^{r}<\mathrm{âˆž}$, so ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}{a}_{ni}{X}_{ni}$ a.s. converges for all $nâ‰¥1$. If $1â‰¤r<2$, by Theorem 2.17 of Hall and Heyde [15], we can get that ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}{a}_{ni}\left({X}_{ni}âˆ’{b}_{ni}\right)$ a.s. converges for all $nâ‰¥1$. Thus ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}{a}_{ni}\left({X}_{ni}âˆ’{b}_{ni}\right)$ a.s. converges for all $nâ‰¥1$ in the case of ${u}_{n}=âˆ’\mathrm{âˆž}$ and/or ${v}_{n}=+\mathrm{âˆž}$.

Let ${X}_{ni}^{â€²}={X}_{ni}I\left(|{X}_{ni}|â‰¤h\left(n\right)\right)âˆ’h\left(n\right)I\left({X}_{ni}<âˆ’h\left(n\right)\right)+h\left(n\right)I\left({X}_{ni}>h\left(n\right)\right)$ and ${X}_{ni}^{â€³}={X}_{ni}âˆ’{X}_{ni}^{â€²}=\left({X}_{ni}+h\left(n\right)\right)I\left({X}_{ni}<âˆ’h\left(n\right)\right)+\left({X}_{ni}âˆ’h\left(n\right)\right)I\left({X}_{ni}>h\left(n\right)\right)$ for ${u}_{n}â‰¤iâ‰¤{u}_{n}$, $nâ‰¥1$.

Case $0. By the ${C}_{r}$-inequality, we obtain

$E{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}{X}_{ni}|}^{r}â‰¤E{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}{X}_{ni}^{â€²}|}^{r}+E{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}{X}_{ni}^{â€³}|}^{r}.$

Noting that $|{X}_{ni}^{â€³}|â‰¤\left(|{X}_{ni}|âˆ’h\left(n\right)\right)I\left(|{X}_{ni}|>h\left(n\right)\right)$ for all ${u}_{n}â‰¤iâ‰¤{v}_{n}$, $nâ‰¥1$, by the ${C}_{r}$-inequality, we obtain

$\begin{array}{rcl}E{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}{X}_{ni}^{â€³}|}^{r}& â‰¤& \underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}^{â€³}|}^{r}\\ â‰¤& \underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{\left(|{X}_{ni}|âˆ’h\left(n\right)\right)}^{r}I\left(|{X}_{ni}|>h\left(n\right)\right)â†’0.\end{array}$

Since $|{X}_{ni}^{â€²}|â‰¤min\left\{|{X}_{ni}|,h\left(n\right)\right\}$ and $0, we have

$\begin{array}{rcl}E|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}{X}_{ni}^{â€²}|& â‰¤& \underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}|E|{X}_{ni}^{â€²}|\\ â‰¤& C{\left(h\left(n\right)\underset{{u}_{n}â‰¤iâ‰¤{v}_{n}}{sup}|{a}_{ni}|\right)}^{1âˆ’r}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}|}^{r}â†’0.\end{array}$

So, ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}{a}_{ni}{X}_{ni}^{â€²}â†’0$ in ${L}^{1}$ and hence in ${L}^{r}$. Therefore, the proof is completed when $0.

Case $r=1$. Since

$\begin{array}{r}E|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left[{X}_{ni}âˆ’E\left({X}_{ni}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]|\\ \phantom{\rule{1em}{0ex}}â‰¤E|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left[{X}_{ni}^{â€²}âˆ’E\left({X}_{ni}^{â€²}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]|+E|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left[{X}_{ni}^{â€³}âˆ’E\left({X}_{ni}^{â€³}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]|\\ \phantom{\rule{1em}{0ex}}â‰¤E|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left[{X}_{ni}^{â€²}âˆ’E\left({X}_{ni}^{â€²}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]|+2E\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}{X}_{ni}^{â€³}|.\end{array}$

But

$E\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}{X}_{ni}^{â€³}|â‰¤\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}|E\left(|{X}_{ni}|âˆ’h\left(n\right)\right)I\left(|{X}_{ni}|>h\left(n\right)\right)â†’0.$

By Burkholderâ€™s inequality (Theorem 2.10 of Hall and Heyde [15]), we have

$\begin{array}{r}E{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left[{X}_{ni}^{â€²}âˆ’E\left({X}_{ni}^{â€²}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]|}^{2}\\ \phantom{\rule{1em}{0ex}}â‰¤C\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}E{\left[{X}_{ni}^{â€²}âˆ’E\left({X}_{ni}^{â€²}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]}^{2}\\ \phantom{\rule{1em}{0ex}}â‰¤CE\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}E{|{X}_{ni}^{â€²}|}^{2}\\ \phantom{\rule{1em}{0ex}}â‰¤C\left(h\left(n\right)\underset{{u}_{n}â‰¤iâ‰¤{v}_{n}}{sup}|{a}_{ni}|\right)\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}|{a}_{ni}|E|{X}_{ni}|â†’0,\end{array}$

hereinafter, C always stands for a positive constant not depending on n which may differ from one place to another, thus ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}{a}_{ni}\left[{X}_{ni}^{â€²}âˆ’E\left({X}_{ni}^{â€²}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]â†’0$ in ${L}^{2}$ and hence in ${L}^{1}$. Therefore, the proof is completed when $r=1$.

Case $1. By Burkholderâ€™s inequality, the ${C}_{r}$-inequality and Jensenâ€™s inequality, we have

$\begin{array}{r}E{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left[{X}_{ni}âˆ’E\left({X}_{ni}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]|}^{r}\\ \phantom{\rule{1em}{0ex}}â‰¤CE{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}{\left[{X}_{ni}âˆ’E\left({X}_{ni}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]}^{2}|}^{r/2}\\ \phantom{\rule{1em}{0ex}}â‰¤CE{|\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}{\left[{X}_{ni}^{â€²}âˆ’E\left({X}_{ni}^{â€²}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]}^{2}|}^{r/2}\\ \phantom{\rule{2em}{0ex}}+CE\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}{|\left[{X}_{ni}^{â€³}âˆ’E\left({X}_{ni}^{â€³}|{\mathrm{â„‘}}_{n,iâˆ’1}\right)\right]|}^{r}\\ \phantom{\rule{1em}{0ex}}â‰¤C{\left\{\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}E{|{X}_{ni}^{â€²}|}^{2}\right\}}^{r/2}+C\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}E{|{a}_{ni}{X}_{ni}^{â€³}|}^{r}.\end{array}$

But

$\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}E{|{X}_{ni}^{â€²}|}^{2}â‰¤{\left(h\left(n\right)\underset{{u}_{n}â‰¤iâ‰¤{v}_{n}}{sup}|{a}_{ni}|\right)}^{2âˆ’r}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}|}^{r}â†’0,$

and

$\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}E{|{a}_{ni}{X}_{ni}^{â€³}|}^{r}â‰¤\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{\left(|{X}_{ni}|âˆ’h\left(n\right)\right)}^{r}I\left(|{X}_{ni}|>h\left(n\right)\right)â†’0.$

Therefore, the proof is completed when $1.â€ƒâ–¡

Remark 3.1

1. (i)

Putting ${a}_{ni}={k}_{n}^{âˆ’1/r}$, ${u}_{n}â‰¤iâ‰¤{v}_{n}$, $nâ‰¥1$, if $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is an array of h-integrability with exponent r ($0), then it is residually $\left(r,{h}^{1/r}\right)$-integrable concerning the array $\left\{{a}_{ni}\right\}$. Thus, Theorem 3.1 and Corollary 3.1 of Sung and Lisawadi and Volodin [7] can be obtained from Theorem 3.1.

2. (ii)

Let ${a}_{ni}=1/{n}^{r}$, ${u}_{n}=1$, ${v}_{n}=n$, $nâ‰¥1$, $h\left(n\right)={n}^{\mathrm{Î±}}$, $\mathrm{Î±}âˆˆ\left(0,1/r\right)$, similar to that of Remark 1 of OrdÃ³Ã±ez Cabrera and Volodin [6], Theorem 3.1 and 3.2 and Corollary 3.1 of Chandra and Goswami [8] can be obtained from Theorem 3.1.

Theorem 3.2 Let $1â‰¤r<2$ and $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of rowwise pairwise NQD random variables. Let $\left\{{a}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ be an array of constants and $\left\{h\left(n\right),nâ‰¥1\right\}$ an increasing sequence of positive constants with $h\left(n\right)â†‘\mathrm{âˆž}$ as $nâ†‘\mathrm{âˆž}$. Assume that the following conditions hold:

1. (i)

$\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is residually $\left(r,h\right)$-integrable concerning the array $\left\{{a}_{ni}\right\}$.

2. (ii)

$h\left(n\right){sup}_{{u}_{n}â‰¤iâ‰¤{v}_{n}}|{a}_{ni}|â†’0$.

Then

$\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left({X}_{ni}âˆ’E{X}_{ni}\right)â†’0$

in ${L}^{r}$ and hence in probability as $nâ†’\mathrm{âˆž}$.

Proof The proof is similar to that of Theorem 3.1, we can get ${âˆ‘}_{i={u}_{n}}^{{v}_{n}}{a}_{ni}\left({X}_{ni}âˆ’E{X}_{ni}\right)$ a.s. converges for all $nâ‰¥1$ in the case of ${u}_{n}=âˆ’\mathrm{âˆž}$ and/or ${v}_{n}=+\mathrm{âˆž}$. Let ${X}_{ni}^{â€²}$ and ${X}_{ni}^{â€³}$ as in Theorem 3.1. Without loss of generality, we can assume that ${a}_{ni}>0$ for ${u}_{n}â‰¤iâ‰¤{v}_{n}$, $nâ‰¥1$, then $\left\{{a}_{ni}{X}_{ni}^{â€²},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ and $\left\{{a}_{ni}{X}_{ni}^{â€³},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ are arrays of rowwise NQD random variables by Lemma 2.1. Observe that

$\begin{array}{rcl}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left({X}_{ni}âˆ’E{X}_{ni}\right)& =& \underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left({X}_{ni}^{â€²}âˆ’E{X}_{ni}^{â€²}\right)+\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}\left({X}_{ni}^{â€³}âˆ’E{X}_{ni}^{â€³}\right)\\ =:& {A}_{n}+{B}_{n}.\end{array}$

By Lemma 2.3 and $|{X}_{ni}^{â€²}|â‰¤min\left\{|{X}_{ni}|,h\left(n\right)\right\}$, we have

$\begin{array}{rcl}E{\left({A}_{n}\right)}^{2}& â‰¤& C\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}E{\left({X}_{ni}^{â€²}âˆ’E{X}_{ni}^{â€²}\right)}^{2}\\ â‰¤& C\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{a}_{ni}^{2}E{\left({X}_{ni}^{â€²}\right)}^{2}\\ â‰¤& C{\left(h\left(n\right)\underset{{u}_{n}â‰¤iâ‰¤{v}_{n}}{sup}|{a}_{ni}|\right)}^{2âˆ’r}\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}|}^{r}â†’0,\end{array}$

then ${A}_{n}â†’0$ in ${L}^{2}$ and hence in ${L}^{r}$.

By Lemma 2.3 and $|{X}_{ni}^{â€³}|â‰¤\left(|{X}_{ni}|âˆ’h\left(n\right)\right)I\left(|{X}_{ni}|>h\left(n\right)\right)$, we have

$\begin{array}{rcl}E{|{B}_{n}|}^{r}& â‰¤& C\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{\left({X}_{ni}^{â€³}âˆ’E{X}_{ni}^{â€³}\right)}^{r}\\ â‰¤& C\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{|{X}_{ni}^{â€³}|}^{r}\\ â‰¤& C\underset{i={u}_{n}}{\overset{{v}_{n}}{âˆ‘}}{|{a}_{ni}|}^{r}E{\left(|{X}_{ni}|âˆ’h\left(n\right)\right)}^{r}I\left(|{X}_{ni}|>h\left(n\right)\right)â†’0.\end{array}$

Thus, the proof is completed.â€ƒâ–¡

Remark 3.2 Theorem 3.2 extended the result in Chen [16] who first obtained the r-the moment convergence under the r th uniform integrability for pairwise NQD sequence.

Remark 3.3

1. (i)

Let $r=1$, ${a}_{ni}={k}_{n}^{âˆ’1}$, ${u}_{n}â‰¤iâ‰¤{v}_{n}$, $nâ‰¥1$, then Theorem 3.2, Corollary 3.2 of Sung et al. [7] and Theorem 2.2 of Yuan and Tao [9] can be obtained from Theorem 3.2.

2. (ii)

Theorem 2.2 of Chandra and Goswami [8] can be obtained from Theorem 3.2.

3. (iii)

Theorem 1 and Corollary 1 of OrdÃ³Ã±ez Cabrera and Volodin [6] can be obtained from Theorem 3.2.

Remark 3.4 Putting ${a}_{ni}={k}_{n}^{âˆ’1/r}$, ${u}_{n}â‰¤iâ‰¤{v}_{n}$, $nâ‰¥1$, if $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is an array of h-integrability with exponent r ($0), then $\left\{{X}_{ni},{u}_{n}â‰¤iâ‰¤{v}_{n},nâ‰¥1\right\}$ is residually $\left(r,{h}^{1/r}\right)$-integrable concerning the array $\left\{{a}_{ni}\right\}$. Theorem 3.3 and Corollary 3.3 of Sung et al. [7] can be obtained from Theorem 3.2 since an NA sequence is an NQD sequence.

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## Acknowledgements

The authors are very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper.

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Cao, L. Mean convergence theorems for weighted sums of random variables under a condition of weighted integrability. J Inequal Appl 2013, 558 (2013). https://doi.org/10.1186/1029-242X-2013-558