We first recall some concepts of dependent structure. The following famous concept of negative quadrant dependent (NQD) was introduced by Lehmann [1]

Definition 1.1 Two random variables $X$ and $Y$ are said to be NQD if

$\begin{aligned} P(X \leq x, Y\leq y) &\leq P(X\leq x)P(Y\leq y)\quad\mbox{for}\;\mbox{all}\;x\;\mbox{and}\;y. \end{aligned}$

A sequence of random variables $\{X_n, n\geq1\}$ is said to be pairwise NQD if every pair of random variables in the sequence are NQD.

Definition 1.2 The random variables $X_1, \cdots, X_k$ are said to be negatively upper orthant dependent (NUOD) if for all real $x_1, \cdots, x_k$,

$\begin{aligned} P(X_i>x_i, i=1, 2, \cdots, k)&\leq\prod\limits_{i=1}^kP(X_i>x_i), \end{aligned}$

and negatively lower orthant dependent (NLOD) if

$\begin{aligned} P(X_i\leq x_i, i=1, 2, \cdots, k)&\leq\prod\limits_{i=1}^kP(X_i\leq x_i). \end{aligned}$

Random variables $X_1, \cdots, X_k$ are said to be negatively orthant dependent (NOD) if they are both NUOD and NLOD. This concept was introduced by Ebrahimi and Ghosh [2].

Joag-Dev and Proschan [3] introduced the concept of negatively associated (NA).

Definition 1.3 A finite family of random variables $\{X_k, 1\leq k\leq n\}$ is said to be NA if for any disjoint subsets $A$ and $B$ of $\{1, 2, \cdots, n\}$ and any real coordinatewise nondecreasing functions $f$ on $R^A$ and $g$ on $R^B$,

$\begin{aligned} {\text{Cov}}(f(X_i, i \in A), g(X_j, j\in B))&\leq 0, \end{aligned}$

whenever the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA.

The concept of linearly negative quadrant dependent (LNQD) random variables was introduced by Newman [4].

Definition 1.4 A sequence $\{X_n, n\geq1\}$ of random variables is said to be LNQD if for any disjoint subsets $A, B\subset\mathbf{Z^+}$ and positive $r_j^{'}s$, $\sum\limits_{k\in A}r_kX_k$ and $\sum\limits_{j\in B}r_jX_j$ are NQD.

Remark 1.1 It is important to note that NOD, NA or LNQD implies pairwise NQD.

It is well known that sequences of pairwise NQD random variables are a family of very wide scope and have been an attractive research topic in the recent papers. We refer reader to Matula [5], Wu [6], Liang et al. [7], Cabrera and Volodin [8], Wan [9], Chen [10], Li and Yang [11], Baek et al. [12], Gan and Chen [13], Meng and Lin [14], Baek and Park [15], Gan and Chen [16], Wu and Jiang [17], Wu and Guan [18], Wu and Wang [19], Sung [20].

As we know, moment inequalities are very important tools in establishing the limit theorems for sequences of random variables. For pairwise NQD random variables, the following Marcinkiewicz-Zygmund inequality with exponent $2$

$\begin{aligned} E\biggl|\sum\limits_{k=1}^nX_k\biggr|^2 &\leq \sum\limits_{k=1}^nE|X_k|^2 \end{aligned}$

(1.1)

has been proved by Wu [6] (see Lemma 2.2). However, due to the limitation of the exponent 2 in the inequality (1.1), many authors could not obtain desirable results of the convergence properties for pairwise NQD random variables.

According to our knowledge, the following Marcinkiewicz-Zygmund inequality with exponent $p$ ($1<p<2$)

$\begin{aligned} E\biggl|\sum\limits_{k=1}^nX_k\biggr|^p&\leq C\sum\limits_{k=1}^nE|X_k|^p \end{aligned}$

(1.2)

for pairwise NQD random variables has not been discussed in previous literature. It is not in doubt that the inequality (1.2) has stronger application value. In this article, we will prove the above inequality (1.2) remains true for pairwise NQD random variables.

Recently Sung ^[20] obtained a $L^r$ convergence result for weighted sums of arrays of rowwise pairwise NQD random variables.

Theorem A Let $\{X_{ni}, u_n\leq i\leq v_n, n \geq1\}$ be an array of rowwise pairwise NQD random variables and $1\leq r<2$. Let $\{a_{ni}, u_n\leq i\leq v_n, n \geq1\}$ be an array of constants. Suppose that

(ⅰ)$\;\;$$\sup\limits_{n\geq1}\sum\limits_{i=u_n}^{v_n}|a_{ni}|^rE|X_{ni}|^r <\infty$,

(ⅱ) $\sum\limits_{i=u_n}^{v_n} |a_{ni}|^rE|X_{ni}|^rI(|a_{ni}|^r|X_{ni}|^r >\varepsilon) \rightarrow0 \;\mathrm{as}\; n\rightarrow\infty$ for any$ \varepsilon>0 $Then

$\begin{aligned} \sum\limits_{i=u_n}^{v_n}a_{ni}(X_{ni}-EX_{ni})\rightarrow0 \end{aligned}$

in $L^r$ and, hence, in probability as $n\rightarrow\infty$.

Chen [10] presented the following $L^r$ convergence result for sequence of pairwise NQD random variables.

Theorem B Let $\{X_n, \, n\geq1\}$ be a sequence of pairwise NQD random variables with $EX_n=0$ for all $n\geq1$ and $1\leq r<2$. $S_n=\sum\limits_{i=1}^nX_i$. Suppose that

$\begin{aligned} \lim\limits_{a\rightarrow\infty}\sup\limits_{n\geq1}n^{-1}\sum\limits_{i=1}^nE|X_i|^rI(|X_i|>a)=0 \end{aligned}$

(1.3)

then $n^{-1/r}S_n\rightarrow0$ in $L^r$ as $n\rightarrow\infty$.

In this work, we first establish the Marcinkiewicz-Zygmund inequality with exponent $p$ ($1<p<2$) for pairwise NQD random variables. As applications, Theorem A and Theorem B are proved by some methods which are much simpler than those in Chen [10] and Sung [20].

Throughout this paper, the symbol $C$ represents positive constants whose values may change from one place to another.

To prove our main result, we need the following technical lemmas.

Lemma 2.1 (see [1]) Let $\{X_n, n\geq1\}$ be a sequence of pairwise NQD random variables. Let $\{f_n, n\geq1\}$ be a sequence of increasing functions. Then $\{f_n(X_n), n\geq1\}$ is a sequence of pairwise NQD random variables.

Lemma 2.2 (see [6]) Let $\{X_n, n\geq1\}$ be a sequence of pairwise NQD random variables with mean zero and $EX_n^2<\infty$. Then $\begin{aligned} E\biggl|\sum_{k=1}^nX_k\biggr|^2\leq \sum_{k=1}^nE|X_k|^2. \end{aligned}$

Now we present the Marcinkiewicz-Zygmund inequality with exponent $p$ ($1<p<2$) for pairwise NQD random variables.

Theorem 2.1 Let $\{X_n, n\geq1\}$ be a sequence of pairwise NQD random variables with mean zero and $E|X_n|^p<\infty$ for $1\leq p<2$. Then there exists a positive constant $C$ depending only on $p$, such that

$\begin{aligned} E\biggl|\sum\limits_{k=1}^nX_k\biggr|^p\leq C\sum\limits_{k=1}^nE|X_k|^p. \end{aligned}$

(2.1)

Proof Let $\varphi_n=\sum\limits_{k=1}^nE|X_k|^p$. For all $t\geq 1$, let

$\begin{aligned} Y_{k}&= -\varphi_n^{1/p}t^{1/p} I(X_{k}<-\varphi_n^{1/p} t^{1/p})+X_{k}I(|X_{k}|\leq\varphi_n^{1/p} t^{1/p})+\varphi_n^{1/p}t^{1/p}I(X_{k}>\varphi_n^{1/p}t^{1/p}), \\ Z_{k}&= X_{k}-Y_{k}\;=\; (X_{k}+\varphi_n^{1/p}t^{1/p}) I(X_{k}<-\varphi_n^{1/p}t^{1/p})+ (X_{k}-\varphi_n^{1/p}t^{1/p}) I(X_{k}>\varphi_n^{1/p}t^{1/p}). \end{aligned}$

By Lemma 2.1, it follows that {$Y_k, k\ge 1\}$ and $\{Z_k, k\ge 1\}$ are sequences of pairwise NQD random variables. Then

$\begin{aligned} &E\biggl|3^{-1}\varphi_n^{-1/p}\sum\limits_{k=1}^nX_k\biggr|^p\\ &= \int_{0}^{\infty} P\biggl(\biggl|\sum\limits_{k=1}^{n}X_k\biggr|\geq 3\, \varphi_n^{1/p}\, t^{1/p}\biggl){\mathrm{d}}\;\leq\; 1+\int_{1}^{\infty}P\biggl(\biggl|\sum\limits_{k=1}^{n}X_k\biggr|\geq 3\, \varphi_n^{1/p}\, t^{1/p}\biggl){\mathrm{d}}t\\ &\leq 1+\sum\limits_{k=1}^n\int_{1}^{\infty}P\bigl(|X_k|>\varphi_n^{1/p} t^{1/p}\bigr){\mathrm{d}}t+\int_{1}^{\infty} P\biggl(\biggl|\sum\limits_{k=1}^{n}Y_k\biggr|\geq 3\, \varphi_n^{1/p}\, t^{1/p}\biggl){\mathrm{d}}t\\ &=\;:1+I_1+I_2. \end{aligned}$

Noting that $\int_1^{\infty}P\bigl(|X_k|>\varphi_n^{1/p}t^{1/p}\bigr){\mathrm{d}}t\leq \varphi_n^{-1}E|X_{k}|^pI(|X_{k}|>\varphi_n^{1/p})$. Hence,

$\begin{aligned} I_1&\leq\varphi_n^{-1}\sum\limits_{k=1}^nE|X_{k}|^pI(|X_{k}|>\varphi_n^{1/p})\;\leq\;1. \end{aligned}$

By $EX_k=0$ and $p\geq1$, we have

$\begin{aligned} &\sup\limits_{t\geq 1}\varphi_n^{-1/p}t^{-1/p}\biggl|\sum\limits_{k=1}^{n}EY_k\biggr|\\ &=\sup\limits_{t\geq 1}\varphi_n^{-1/p}t^{-1/p}\biggl|\sum\limits_{k=1}^{n}\bigl\{-\varphi_n^{1/p}t^{1/p} P(X_{k}<-\varphi_n^{1/p} t^{1/p})\\ &\qquad+EX_{k}I(|X_{k}|\leq\varphi_n^{1/p} t^{1/p})+\varphi_n^{1/p}t^{1/p}P(X_{k}>\varphi_n^{1/p}t^{1/p})\bigr\}\biggr|\\ &=\sup\limits_{t\geq 1}\varphi_n^{-1/p}t^{-1/p}\biggl|\sum\limits_{k=1}^{n}\bigl\{-\varphi_n^{1/p}t^{1/p} P(X_{k}<-\varphi_n^{1/p} t^{1/p})\\ &\qquad-EX_{k}I(|X_{k}|>\varphi_n^{1/p} t^{1/p})+\varphi_n^{1/p}t^{1/p}P(X_{k}>\varphi_n^{1/p}t^{1/p})\bigr\}\biggr|\\ &\leq\sup\limits_{t\geq 1}\varphi_n^{-1/p}t^{-1/p}\sum\limits_{k=1}^{n}\bigl\{\varphi_n^{1/p}t^{1/p} P(|X_{k}|>\varphi_n^{1/p} t^{1/p})+E|X_{k}|I(|X_{k}|>\varphi_n^{1/p} t^{1/p})\bigr\}\\ &\quad\leq\sup\limits_{t\geq 1}\sum\limits_{k=1}^{n}P(|X_{k}|>\varphi_n^{1/p} t^{1/p})+\sup\limits_{t\geq 1}\varphi_n^{-1}t^{-1}\sum\limits_{k=1}^{n}E|X_{k}|^pI(|X_{k}|>\varphi_n^{1/p} t^{1/p})\\ &\quad\leq2\varphi_n^{-1}\sum\limits_{k=1}^{n}E|X_{k}|^pI(|X_{k}|>\varphi_n^{1/p})\leq 2. \end{aligned}$

Therefore, $\bigl|\sum\limits_{k=1}^{n}EY_k\bigr|\leq2\, \varphi_n^{1/p}t^{1/p}$ holds uniformly for $t\geq 1$. Then

$\begin{aligned} I_2&\leq\int_{1}^{\infty}P\biggl(\biggl|\sum\limits_{k=1}^{n}(Y_k-EY_k)\biggr|\geq\varphi_n^{1/p}t^{1/p}\biggl){\mathrm{d}}t. \end{aligned}$

By the Markov inequality, Lemma 2.2 and $C_r$-inequality, we have

$\begin{aligned} I_2&\leq\varphi_n^{-2/p}\int_{1}^{\infty}t^{-2/p}E\biggl|\sum\limits_{k=1}^{n}(Y_k-EY_k)\biggr|^2{\mathrm{d}}t\\ &\leq\varphi_n^{-2/p}\sum\limits_{k=1}^{n}\int_{1}^{\infty}t^{-2/p}E(Y_k-EY_k)^2{\mathrm{d}}t \;\leq\;2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\int_{1}^{\infty}t^{-2/p}EY_k^2{\mathrm{d}}t\\ &=2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\int_{1}^{\infty}t^{-2/p}EX_k^2I(|X_{k}|\leq\varphi_n^{1/p}t^{1/p}){\mathrm{d}}t +2\sum\limits_{k=1}^{n}\int_{1}^{\infty}P(|X_{k}|>\varphi_n^{1/p}t^{1/p}){\mathrm{d}}t\\ &=2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\int_{1}^{\infty}t^{-2/p}EX_k^2I(|X_{k}|\leq\varphi_n^{1/p}){\mathrm{d}}t +2\sum\limits_{k=1}^{n}\int_{1}^{\infty}P(|X_{k}|>\varphi_n^{1/p}t^{1/p}){\mathrm{d}}t\\ &+2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\int_{1}^{\infty}t^{-2/p}EX_k^2I(\varphi_n^{1/p}<|X_{k}|\leq\varphi_n^{1/p}t^{1/p}){\mathrm{d}}t\\ &=\;:I_3+I_4+I_5. \end{aligned}$

By a similar argument as in the proof of $I_1\leq1$, we can prove $I_4\leq 2$. By $p<2$, we get

$\begin{aligned} I_3&=\frac{2p}{2-p}\varphi_n^{-2/p}\sum\limits_{k=1}^{n}EX_k^2I(|X_{k}|\leq \varphi_n^{1/p})\\ &\leq\frac{2p}{2-p}\varphi_n^{-1}\sum\limits_{k=1}^{n}E|X_k|^pI(|X_{k}|\leq \varphi_n^{1/p})\;\leq\;\frac{2p}{2-p}. \end{aligned}$

Finally we estimate $I_5$. Noting that $\sum\limits_{m=s}^{\infty}m^{-2/p}\leq2/(2-p)s^{1-2/p}$ and $(s+1)/s\leq2$ for all $s\geq1$. We can get

$\begin{aligned} I_5&=2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\sum\limits_{m=1}^{\infty}\int_{m}^{m+1}t^{-2/p}EX_k^2 I(\varphi_n^{1/p}<|X_{k}|\leq\varphi_n^{1/p}t^{1/p}){\mathrm{d}}t\\ &\leq2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\sum\limits_{m=1}^{\infty}m^{-2/p}EX_k^2 I(\varphi_n<|X_{k}|^p\leq\varphi_n(m+1))\\ &=2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\sum\limits_{m=1}^{\infty}m^{-2/p}\sum\limits_{s=1}^mEX_k^2 I(\varphi_n\, s<|X_{k}|^p\leq\varphi_n(s+1))\\ &=2\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\sum\limits_{s=1}^{\infty}EX_k^2 I(\varphi_n\, s<|X_{k}|^p\leq\varphi_n(s+1))\sum\limits_{m=s}^{\infty}m^{-2/p}\\ &\leq\frac{4}{2-p}\, \varphi_n^{-2/p}\sum\limits_{k=1}^{n}\sum\limits_{s=1}^{\infty}s^{1-2/p}EX_k^2 I(\varphi_n\, s<|X_{k}|^p\leq\varphi_n(s+1))\\ &\leq\frac{2^{2/p+1}}{2-p}\, \varphi_n^{-1}\sum\limits_{k=1}^{n}\sum\limits_{s=1}^{\infty}E|X_k|^p I(\varphi_n\, s<|X_{k}|^p\leq\varphi_n(s+1))\\ &=\frac{2^{2/p+1}}{2-p}\, \varphi_n^{-1}\sum\limits_{k=1}^{n}E|X_k|^p I(|X_{k}|^p>\varphi_n)\;\leq\;\frac{2^{2/p+1}}{2-p}. \end{aligned}$

From $I_1\leq1$, $I_3\leq 2\, p/(2-p)$, $I_4\leq 2$ and $I_5\leq\, \frac{2^{2/p+1}}{2-p}$, we have

$\begin{aligned} E\biggl|3^{-1}\varphi_n^{-1/p}\sum\limits_{k=1}^nX_k\biggr|^p\;\leq\;4+\frac{2\, p}{2-p}+\frac{2^{2/p+1}}{2-p}. \end{aligned}$

Let $C=3^p\bigl(4+\frac{2\, p}{2-p}+\frac{2^{2/p+1}}{2-p}\bigr)$. Clearly $C$ depends only on $p$, then we get

$\begin{aligned} E\biggl|\sum\limits_{k=1}^nX_k\biggr|^p\;\leq\;C\, \sum\limits_{k=1}^nE|X_k|^p. \end{aligned}$

The proof is completed.

Remark 2.1  The inequality (2.1) for NA and NOD random variables has been proved by Shao [21] and Asadian et al. [22], respectively. The above result shows that the famous Marcinkiewicz-Zygmund inequality with exponent $p$ ($1\leq p<2$) remains true for pairwise NQD random variables.

As applications, we prove Theorem A and Theorem B by some simpler methods compared with Chen [10] and Sung [20].

Proof of Theorem A Without loss of generality, we may assume that $a_{ni}\geq0$. For $u_n\leq i\leq v_n$, $n\geq1$, let

$\begin{aligned} Y_{ni}&=-\varepsilon^{1/r} I(a_{ni}X_{ni}<-\varepsilon^{1/r})+a_{ni}X_{ni} I(a_{ni}|X_{ni}|\leq \varepsilon^{1/r})+\varepsilon^{1/r} I(a_{ni}X_{ni}>\varepsilon^{1/r}), \\ Z_{ni}&=a_{ni}X_{ni}-Y_{ni}. \end{aligned}$

By Lemma 2.1, $\{Y_{ni}, u_n\leq i\leq v_n, n \geq1\}$ and $\{Z_{ni}, u_n\leq i\leq v_n, n \geq1\}$ are arrays of rowwise pairwise NQD. Given $\varepsilon>0$, by Theorem 2.1, we have

$\begin{aligned} &E\Biggl|\sum\limits_{i=u_n}^{v_n}a_{ni}(X_{ni}-EX_{ni})\Biggr|^r\\ &\leq2^{r-1}\Biggl\{E\Biggl|\sum\limits_{i=u_n}^{v_n}(Z_{ni}-EZ_{ni})\Biggr|^r +E\Biggl|\sum\limits_{i=u_n}^{v_n}(Y_{ni}-EY_{ni})\Biggr|^r\Biggr\}\\ &\leq2^{r-1}E\Biggl|\sum\limits_{i=u_n}^{v_n}(Z_{ni}-EZ_{ni})\Biggr|^r +2^{r-1}\Biggl\{E\Biggl|\sum\limits_{i=u_n}^{v_n}(Y_{ni}-EY_{ni})\Biggr|^2\Biggr\}^{r/2}\\ &\leq C\, 2^{r-1}\sum\limits_{i=u_n}^{v_n}E|Z_{ni}|^r+C\, 2^{r-1}\Biggl\{\sum\limits_{i=u_n}^{v_n}EY_{ni}^2\Biggr\}^{r/2}\\ &=\;:I_6+I_7. \end{aligned}$

Noting that $|Z_{ni}|\leq a_{ni}|X_{ni}|I(a_{ni}^r|X_{ni}|^r>\varepsilon)$. By the condition (ⅱ), we have

$\begin{aligned} I_6\;\leq\;C\, \sum\limits_{i=u_n}^{v_n}a_{ni}^rE|X_{ni}|^rI(a_{ni}^r|X_{ni}|^r>\varepsilon)\rightarrow0\quad\mbox{as}\;\;n\rightarrow\infty. \end{aligned}$

Next we prove $I_7\rightarrow0$ as $n\rightarrow\infty$. Without loss of generality, we may assume $0<\varepsilon<1$. Then

$\begin{aligned} I_7^{2/r}&\leq C\sum\limits_{i=u_n}^{v_n}a_{ni}^2EX_{ni}^2I(a_{ni}^r|X_{ni}|^r\leq\varepsilon) +C\varepsilon^{2/r}\sum\limits_{i=u_n}^{v_n}P(a_{ni}^r|X_{ni}|^r>\varepsilon)\\ &\leq C\sum\limits_{i=u_n}^{v_n}a_{ni}^2EX_{ni}^2I(a_{ni}^r|X_{ni}|^r\leq\varepsilon^2) +C\sum\limits_{i=u_n}^{v_n}a_{ni}^2EX_{ni}^2I(\varepsilon^2<a_{ni}^r|X_{ni}|^r\leq\varepsilon)\\ &\qquad+C\varepsilon^{2/r-1}\sum\limits_{i=u_n}^{v_n}a_{ni}^rE|X_{ni}|^rI(a_{ni}^r|X_{ni}|^r>\varepsilon)\\ &=\;: I_8+I_9+I_{10}. \end{aligned}$

By $r<2$ and (ⅱ), we get $I_{10}\rightarrow0$ as $n\rightarrow\infty$. For $I_8$, we have

$\begin{aligned} I_8&\leq C\, \varepsilon^{4/r-2}\sum\limits_{i=u_n}^{v_n}a_{ni}^rE|X_{ni}|^rI(a_{ni}^r|X_{ni}|^r\leq\varepsilon^2)\\ &\leq C\, \varepsilon^{4/r-2}\sup\limits_{n\geq1}\sum\limits_{i=u_n}^{v_n}a_{ni}^rE|X_{ni}|^r. \end{aligned}$

By $r<2$ and (ⅱ), we have

$\begin{aligned} I_9&\leq C\, \varepsilon^{2/r-1}\sum\limits_{i=u_n}^{v_n}a_{ni}^rE|X_{ni}|^rI(\varepsilon^2< a_{ni}^r|X_{ni}|^r\leq\varepsilon)\\ &\leq C\, \varepsilon^{2/r-1}\sum\limits_{i=u_n}^{v_n}a_{ni}^rE|X_{ni}|^rI(a_{ni}^r|X_{ni}|^r>\varepsilon^2) \rightarrow0\quad\mbox{as}\;\;n\rightarrow\infty. \end{aligned}$

Therefore,

$\begin{aligned} \lim\limits_{n\rightarrow\infty}\sup E\Biggl|\sum\limits_{i=u_n}^{v_n}a_{ni}(X_{ni}-EX_{ni})\Biggr|^r\, \leq\, C\, \varepsilon^{2-r} \biggl(\sup\limits_{n\geq1}\sum\limits_{i=u_n}^{v_n}a_{ni}^rE|X_{ni}|^r\biggr)^{r/2}. \end{aligned}$

Since $0<\varepsilon<1$ is arbitrary, by $r<2$ and (ⅰ), the proof is completed.

Proof of Theorem B Let $a=n^{(1-r/2)/4}$ and

$\begin{aligned} Y_{i}&= -aI(X_{i}<-a)+X_{i}I(|X_{i}|\leq a)+aI(X_{i}>a), \\ Z_{i}&= X_{i}-Y_{i}\, =\, (X_{i}+a)I(X_{i}<-a)+(X_{i}-a)I(X_{i}>a). \end{aligned}$

By Lemma 2.1, $\{Y_{i}, i\geq1\}$ and $\{Z_{i}, i\geq1\}$ are sequences of pairwise NQD. By Theorem 2.1 and Lemma 2.2, we have

$\begin{aligned} E\biggl|n^{-1/r}\sum\limits_{i=1}^{n}X_{i}\biggr|^r &\leq Cn^{-1}\Biggl\{E\biggl|\sum\limits_{i=1}^{n}(Z_{i}-EZ_{i})\biggr|^r +E\biggl|\sum\limits_{i=1}^{n}(Y_{i}-EY_{i})\biggr|^r\Biggr\}\\ &\leq Cn^{-1}E\biggl|\sum\limits_{i=1}^{n}(Z_{i}-EZ_{i})\biggr|^r +Cn^{-1}\Biggl\{E\biggl|\sum\limits_{i=1}^{n}(Y_{i}-EY_{i})\biggr|^2\Biggr\}^{r/2}\\ &\leq Cn^{-1}\sum\limits_{i=1}^{n}E|Z_{i}|^r+Cn^{-1}\Biggl\{\sum\limits_{i=1}^{n}EY_{i}^2\Biggr\}^{r/2}\\ &=\;: I_{11}+I_{12}. \end{aligned}$

By $|Z_{i}|\leq |X_{i}|I(|X_{i}|>a)$ and (1.3), we have

$\begin{aligned} I_{11}\;\leq\;C\sup\limits_{n\geq1}n^{-1}\sum\limits_{i=1}^{n}E|X_{i}|^rI(|X_{i}|>a)\rightarrow0\quad\mbox{as}\;\;n\rightarrow\infty. \end{aligned}$

By $|Y_{i}|\leq a=n^{(1-r/2)/4}$ and $r<2$, we have

$\begin{aligned} I_{12}\;\leq\;Cn^{-1}\Biggl\{\sum\limits_{i=1}^{n}a^2\Biggr\}^{r/2}=Cn^{(2-r)(r-4)/8}\rightarrow0\quad\mbox{as}\;\;n\rightarrow\infty. \end{aligned}$

The proof is completed.