© Wuhan University 2024

0 Introduction

In many statistical models, it is unrealistic to assume that random variables are independent, so many scholars have extended the independent random variables to dependent random variables, for example, negatively associated (NA) random variables, negatively orthant dependent (NOD) random variables, and extend negatively dependent (END) random variables and so on.

The widely orthant dependent (WOD) random variables sequences are very important dependent sequences, which was introduced by Wang et al^[1].

Definition 1   The random variables $\left\{X_n,n\ge \mathrm{1}\right\}$ are called to be widely upper orthant dependent (WUOD) random variables, if there exists a finite sequence of real numbers $\left\{g_U(n),n\ge \mathrm{1}\right\}$ such that for each $n\ge \mathrm{1}$, $x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_n\in \mathbb{R}$,

$P(X_{\mathrm{1}}>x_{\mathrm{1}},X_{\mathrm{2}}>x_{\mathrm{2}},\cdots ,X_n>x_n)\le g_U(n){\displaystyle \prod _{i=\mathrm{1}}^n}P(X_i>x_i).$(1)

The random variables $\left\{X_n,n\ge \mathrm{1}\right\}$ are called to be widely lower orthant dependent (WLOD) random variables, if there exists a finite sequence of real numbers $\left\{g_L(n),n\ge \mathrm{1}\right\}$ such that for each $x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_n\in \mathbb{R}$,

$P(X_{\mathrm{1}}\le x_{\mathrm{1}},X_{\mathrm{2}}\le x_{\mathrm{2}},\cdots ,X_n\le x_n)\le g_L(n){\displaystyle \prod _{i=\mathrm{1}}^n}P(X_i\le x_i).$(2)

The random variables $\left\{X_n,n\ge \mathrm{1}\right\}$ are called to be WOD random variables, if the random variables $\left\{X_n,n\ge \mathrm{1}\right\}$ are both WUOD and WLOD, $g(n)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{g_U(n),g_L(n)\right\}$ are called dominated coefficients.

Inspired by m-NA and WOD, the concept of m-WOD random variables was introduced by Fang et al^[2], as follows:

Definition 2   For fix integer $m\ge \mathrm{1}$, the random variables $\left\{X_n,n\ge \mathrm{1}\right\}$ is called to be m-WOD if for any $n\ge \mathrm{2}$, $i_{\mathrm{1}},i_{\mathrm{2}},\cdots ,i_n\in {\mathbb{N}}^{+}$, such that $|i_k-i_j|\ge m$ for all $\mathrm{1}\le k\ne j\le n$, the $X_{i_{\mathrm{1}}},X_{i_{\mathrm{2}}},\cdots ,X_{i_n}$ also are WOD random variables.

From (1) and (2), we have $g_U(n)\ge \mathrm{1},g_L(n)\ge \mathrm{1}$. If $g_U(n)=g_L(n)=\mathrm{1}$, then WOD random variables are NOD random variables, which were introduced by Ebrahimi and Ghosh^[3]. If $g_U(n)=g_L(n)=M\ge \mathrm{1}$, then WOD random variables are END random variables, which were introduced by Liu^[4]. Liu^[4] pointed out the END random variables imply NA and positive random variables, so m-WOD random variables include independent random variables, WOD, m-NA, m-NOD, m-END random variables and so on. Therefore, it is interesting to investigate complete convergence of the maximal partial sums for m-WOD random variables.

The real number sequences $\left\{a_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ satisfy ${\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|<\mathrm{\infty }$, the $\left\{X_n,n\ge \mathrm{1}\right\}$ are called to be moving average process under random variables sequences $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$, if

$X_n={\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{Y}_{n+i},n\ge \mathrm{1}.$(3)

Since the moving average process was proposed,many results about convergence properties have been obtained. When $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ is identically distributed, many results have been gained, for example^[5-8]. Recently, some results have been obtained under the sequence $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ are dependent. For example, Li et al^[9] investigated the convergence properties under $\rho$-mixing assumptions; Zhang^[10] and Chen et al^[11] established complete convergence under $\phi$-mixing assumptions; Song and Zhu^[12] got the complete convergence of moving average process based on ${\rho }^{-}$-mixing assumptions; Tao et al^[13] discussed the complete convergence under WOD random variables; Guan et al^[14] obtained the complete moment convergence under m-WOD random variables.

In this paper, based on the research of Guan et al^[14], we study the complete cmoment convergence of moving average processes based on m-WOD random variables, the results extend and improve the corresponding ones under WOD, m-NA, m-NOD, m-END random variables.

Definition 3   The random variables $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ are called to be stochastically dominated by a random variable Y, if for any $x>\mathrm{0}$,

$P(|Y_n|>x)\le CP(|Y|>x),-\mathrm{\infty }<n<\mathrm{\infty },$

where the constant $C>\mathrm{0}$, and denote $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}\prec Y.$

In this paper,$I(A)$ denotes the indicator function of an event $A$, $\mathrm{l}\mathrm{o}\mathrm{g}n=\mathrm{l}\mathrm{n}\mathrm{m}\mathrm{a}\mathrm{x}\left\{x,e\right\},X^{+}=XI(X>\mathrm{0}),g(n)=$$\mathrm{m}\mathrm{a}\mathrm{x}\left\{g_U(n),g_L(n)\right\}.$ The symbol $C$ represents a positive constant, which can take different values in different places, even in the same formula.

1 Some Lemmas and Main Results

Lemma 1   (Fang^[2]) The sequence $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ are m-WOD random variables, if the function $\left\{f_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ are non-decreasing(non-increasing), then $\left\{f_n(Y_n),-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ are also m-WOD random variables sequences with same dominating coefficients.

Lemma 2   (Fang^[2]) The sequence $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ are m-WOD random variables with dominating coefficients $g(n)$. For every $j\ge \mathrm{1}$, the $E{Y}_j=\mathrm{0}$ and $E|Y_j{|}^p<\mathrm{\infty }$. Then, there exist positive constants $C_{\mathrm{1}}=C_{\mathrm{1}}(p,m),C_{\mathrm{2}}=C_{\mathrm{2}}(p,m),$depending only on $p$ and $m$, such that

$\begin{array}{l}E(|{\displaystyle \sum _{i=\mathrm{1}}^n}{Y}_i{|}^p)\le [C_{\mathrm{1}}(p,m)+C_{\mathrm{2}}(p,m)g(n)]{\displaystyle \sum _{i=\mathrm{1}}^n}E|Y_i{|}^p,\mathrm{1}<p\le \mathrm{2};\\ E(|{\displaystyle \sum _{i=\mathrm{1}}^n}{Y}_i{|}^p)\le C_{\mathrm{1}}(p,m){\displaystyle \sum _{i=\mathrm{1}}^n}E|Y_i{|}^p+C_{\mathrm{2}}(p,m)g(n){({\displaystyle \sum _{i=\mathrm{1}}^n}E{Y}_i^{\mathrm{2}})}^{\frac{p}{\mathrm{2}}},p>\mathrm{2}.\end{array}$

Lemma 3   (Fang^[2]) The sequence $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ are m-WOD random variables with dominating coefficients $g(n)$. For every $j\ge \mathrm{1}$, the $E{Y}_j=\mathrm{0}$ and $E|Y_j{|}^p<\mathrm{\infty }$. Then, there exist positive constants $C_{\mathrm{1}}=C_{\mathrm{1}}(p,m),C_{\mathrm{2}}=C_{\mathrm{2}}(p,m),$ depending only on $p$ and $m$, such that

$E(|\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=\mathrm{1}}^k}{Y}_i{|}^p)\le [C_{\mathrm{1}}(p,m)+C_{\mathrm{2}}(p,m)g(n)]\mathrm{l}\mathrm{o}{\mathrm{g}}^{p}n{\displaystyle \sum _{i=\mathrm{1}}^n}E|Y_i{|}^p,\mathrm{1}<p\le \mathrm{2};$

$E(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=\mathrm{1}}^k}{Y}_i{|}^p)\le C_{\mathrm{1}}(p,m)\mathrm{l}\mathrm{o}{\mathrm{g}}^{p}n{\displaystyle \sum _{i=\mathrm{1}}^n}E|Y_i{|}^p+C_{\mathrm{2}}(p,m)g(n)\mathrm{l}\mathrm{o}{\mathrm{g}}^{p}n{({\displaystyle \sum _{i=\mathrm{1}}^n}E{Y}_i^{\mathrm{2}})}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.},p>\mathrm{2}.$

Lemma 4   (Wu^[15]) Constant $a>\mathrm{0},b>\mathrm{0}$, $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}\prec Y$, then there exist positive constants $C_{\mathrm{1}},C_{\mathrm{2}}$ such that following inequalities are established:

$E|Y_n{|}^{a}I(|Y_n|\le b)\le C[E|Y{|}^{a}I(|Y|\le b)+b^{a}I(|Y|>b)],E|Y_n{|}^{a}I(|Y_n|>b)\le CE|Y{|}^{a}I(|Y|>b).$

Lemma 5   (Wu^[16]) Let $\left\{X_n,n\ge \mathrm{1}\right\}$ and $\left\{Y_n,n\ge \mathrm{1}\right\}$ be a sequence of random variable, for any $q>r>\mathrm{0},\epsilon >\mathrm{0},a>\mathrm{0}$, then

$E{(|{\displaystyle \sum _{i=\mathrm{1}}^n}(X_i+Y_i)|-\epsilon a)}_{+}^r\le C_r(\frac{\mathrm{1}}{{\epsilon }^q}+\frac{r}{q-r}\frac{r}{q-r})\frac{\mathrm{1}}{a^{q-r}}E(|{\displaystyle \sum _{i=\mathrm{1}}^n}{X}_i{|}^q)+C_{r}E(|{\displaystyle \sum _{i=\mathrm{1}}^n}{Y}_i{|}^r),$

$E{(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=\mathrm{1}}^k}(X_i+Y_i)|-\epsilon a)}_{+}^r\le C_r(\frac{\mathrm{1}}{{\epsilon }^q}+\frac{r}{q-r})\frac{\mathrm{1}}{a^{q-r}}E(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=\mathrm{1}}^k}{X}_i{|}^q)+C_{r}E(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=\mathrm{1}}^k}{Y}_i{|}^r).$

where $C_r=\mathrm{1}$ if $\mathrm{0}<r\le \mathrm{1}$ or $C_r={\mathrm{2}}^{r-\mathrm{1}}$ if $r>\mathrm{1}$.

Now, we present the main results, the proofs for them will be postponed in next section.

Theorem 1   Let $\gamma >\mathrm{0},\alpha >\mathrm{1}/\mathrm{2},\alpha p>\mathrm{1},$$\left\{X_n,n\ge \mathrm{1}\right\}$ be a moving average process under the sequence $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ of m-WOD random variables with dominating coefficients $g(n)$, $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}\prec Y$, $E|Y|<\mathrm{\infty }$ and $E{Y}_j=\mathrm{0}$ for every $j$. The real number $\left\{a_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ satisfies ${\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|<\mathrm{\infty }$. Then $g(n)=o(n^{\delta })$ for some $\delta \ge \mathrm{0}$, and $\delta <(\gamma /\mathrm{2}-\mathrm{1})(\alpha p-\mathrm{1})$ for $\gamma >\mathrm{2}$. If

$\{\begin{array}{l}E|Y{|}^{p+\delta /\alpha }<\mathrm{\infty },\mathrm{i}\mathrm{f}p>\gamma ,\\ E|Y{|}^{p+\delta /\alpha }\mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{1}+|Y|)<\mathrm{\infty },\mathrm{i}\mathrm{f}p=\gamma ,\\ E|Y{|}^{\gamma +\delta /\alpha }<\mathrm{\infty },\mathrm{i}\mathrm{f}p<\gamma ,\end{array}$

then

${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}{X}_j|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }<\mathrm{\infty },\forall \epsilon >\mathrm{0}.$(4)

and

${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}}P(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}{X}_j|>\epsilon n^{\alpha })<\mathrm{\infty },\forall \epsilon >\mathrm{0}.$(5)

If $\alpha p=\mathrm{1}$, we have the following result:

Theorem 2   Let $\gamma >\mathrm{0},\alpha >\mathrm{1}/\mathrm{2},$$\left\{X_n,n\ge \mathrm{1}\right\}$ be a moving average process under the sequence $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ of m-WOD random variables with dominating coefficients $g(n)$, $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}\prec Y,$$E|Y|<\mathrm{\infty }$ and $E{Y}_j=\mathrm{0}$ for every $j$. The real number $\left\{a_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$ satisfies ${\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|<\mathrm{\infty }$. Then $g(n)=o(n^{\delta })$ for some $\delta \ge \mathrm{0}$, and $\delta <\gamma (\alpha -\mathrm{1}/\mathrm{2})$ for $\gamma >\mathrm{2}$. If

$\{\begin{array}{l}E|Y{|}^{(\mathrm{1}+\delta )/\alpha }<\mathrm{\infty },\mathrm{i}\mathrm{f}\mathrm{1}/\alpha >\gamma ,\\ E|Y{|}^{(\mathrm{1}+\delta )/\alpha }\mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{1}+|Y|)<\mathrm{\infty },\mathrm{i}\mathrm{f}\mathrm{1}/\alpha =\gamma ,\\ E|Y{|}^{\gamma (\mathrm{1}+\delta )}<\mathrm{\infty },\mathrm{i}\mathrm{f}\mathrm{1}/\alpha <\gamma ,\end{array}$

then

${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{-\alpha \gamma -\mathrm{1}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}{X}_j|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }<\mathrm{\infty },\forall \epsilon >\mathrm{0},$(6)

and

${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{-\mathrm{1}}P(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}{X}_j|>\epsilon n^{\alpha })<\mathrm{\infty },\forall \epsilon >\mathrm{0}.$(7)

Remark 1   Theorem 1 and 2 obtain complete $\gamma$-order moment convergence and complete convergence of the maximum partial sums for the moving average process $\left\{X_n,n\ge \mathrm{1}\right\}$, so the results in the paper extend and improve the results in Guan et al^[14] and Song et al^[17].

Remark 2   We known the m-WOD random variables include WOD, m-NA, m-NOD, m-END random variables, and so on, so the results also hold for WOD, m-NA, m-NOD, m-END random variables. The $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}\prec Y$ is weaker than the condition of identical distribution for $\left\{Y_n,-\mathrm{\infty }<n<\mathrm{\infty }\right\}$, therefore Theorem 1 and 2 improve the known results.

2 Proof of Theorems

Proof of Theorem 1   By $E|Y|<\mathrm{\infty }$,${\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|<\mathrm{\infty }$, we get

$E|X_n|\le {\displaystyle \sum _{i=-\infty }^{\infty }}E|a_i{Y}_{n+i}|\le CE|Y|{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|<\mathrm{\infty },\forall n\ge \mathrm{1},$

therefore $X_n$ exist. Set $\frac{\mathrm{1}}{\alpha p}<q<\mathrm{1}$, for $\mathrm{1}\le j\le n,n\ge \mathrm{1}$, let

$\begin{array}{l}{Y}_j^{(n,\mathrm{1})}=-n^{\alpha q}I(Y_j<-n^{\alpha q})+Y_{j}I(|Y_j|\le n^{\alpha q})+n^{\alpha q}I(Y_j>n^{\alpha q}),\\ Y_j^{(n,\mathrm{2})}=(Y_j-n^{\alpha q})I(n^{\alpha q}<Y_j\le n^{\alpha q}+n^{\alpha })+n^{\alpha }I(Y_j>n^{\alpha q}+n^{\alpha }),\\ Y_j^{(n,\mathrm{3})}=(Y_j+n^{\alpha q})I(-n^{\alpha q}-n^{\alpha }<Y_j\le -n^{\alpha q})-n^{\alpha }I(Y_j<-n^{\alpha q}-n^{\alpha }),\\ Y_j^{(n,\mathrm{4})}=(Y_j-n^{\alpha q}-n^{\alpha })I(Y_j>n^{\alpha q}+n^{\alpha }),\\ Y_j^{(n,\mathrm{5})}=(Y_j+n^{\alpha q}+n^{\alpha })I(Y_j<-n^{\alpha q}-n^{\alpha }).\end{array}$

Then

${\displaystyle \sum _{j=\mathrm{1}}^n}{X}_j={\displaystyle \sum _{j=\mathrm{1}}^n}{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{Y}_{j+i}={\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}{\displaystyle \sum _{l=\mathrm{1}}^{\mathrm{5}}}{Y}_j^{(n,l)}.$

From Lemma 4 and $E|Y_j^{(n,\mathrm{2})}|\le E|Y_j|I(|Y_j|>n^{\alpha q})$, we have

$n^{-\alpha }\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|E{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{Y}_j^{(n,\mathrm{2})}|\le n^{-\alpha }{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}E|Y_j^{(n,\mathrm{2})}|\le n^{-\alpha }{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}E|Y_j|I(|Y_j|>n^{\alpha q})\le C{n}^{\mathrm{1}-\alpha }E|Y|I(|Y|>n^{\alpha q}).$(8)

If $\alpha >\mathrm{1}$, we get

$n^{-\alpha }\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|E{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{Y}_j^{(n,\mathrm{2})}|\to \mathrm{0},n\to \mathrm{\infty }.$

If $\alpha \le \mathrm{1}$, for $\frac{\mathrm{1}}{\alpha p}<q<\mathrm{1},\alpha p>\mathrm{1},$ we get $(\mathrm{1}-\alpha pq)-\alpha (\mathrm{1}-q)<\mathrm{0},E|Y{|}^p<\mathrm{\infty },$ we have

$n^{-\alpha }\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|E{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{Y}_j^{(n,\mathrm{2})}|\le C{n}^{\mathrm{1}-\alpha }E|Y|I(|Y|>n^{\alpha q})\le C{n}^{(\mathrm{1}-\alpha pq)-\alpha (\mathrm{1}-q)}E|Y{|}^p\to \mathrm{0},n\to \mathrm{\infty }.$

Similar to the above proof process ,we can obtain

$n^{-\alpha }\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|E{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{Y}_j^{(n,\mathrm{4})}|\le n^{-\alpha }{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}E|Y_j|I(|Y_j|>n^{\alpha q}+n^{\alpha })\le C{n}^{\mathrm{1}-\alpha }E|Y|I(|Y|>n^{\alpha q})\to \mathrm{0},n\to \mathrm{\infty }.$(9)

Similarly, we get

$n^{-\alpha }\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|E{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{Y}_j^{(n,\mathrm{3})}|\to \mathrm{0},n\to \mathrm{\infty }.$(10)

$n^{-\alpha }\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|E{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{Y}_j^{(n,\mathrm{5})}|\to \mathrm{0},n\to \mathrm{\infty }.$(11)

Noting that $E{Y}_j=\mathrm{0}$, it follows from Lemma 5, $C_r$ inequality and (8)-(11), for $v>\gamma \ge \mathrm{1},$

$\begin{array}{l}{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}{X}_j|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }={\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{\displaystyle \sum _{l=\mathrm{1}}^{\mathrm{5}}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)})|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }\\ \le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{{\displaystyle \sum _{l=\mathrm{1}}^{\mathrm{5}}}\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)})|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }\end{array}$

$\begin{array}{l}\le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})})|+{\displaystyle \sum _{l=\mathrm{2}}^{\mathrm{5}}}{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}|a_i||{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}{Y}_j^{(n,l)}|-\frac{\mathrm{3}}{\mathrm{4}}\epsilon n^{\alpha }{\}}_{+}^{\gamma }\\ \le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})})|+{\displaystyle \sum _{l=\mathrm{2}}^{\mathrm{5}}}{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}|a_i||{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)})|-\frac{\mathrm{1}}{\mathrm{2}}\epsilon n^{\alpha }{\}}_{+}^{\gamma }\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})}{)|\}}^v+C{\displaystyle \sum _{l=\mathrm{2}}^{\mathrm{3}}}{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}E\{{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i||{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)}{)|\}}^v\\ {+C{\displaystyle \sum _{l=\mathrm{4}}^{\mathrm{5}}}{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i||{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)})|\}}^{\gamma }\\ =:I{}_{\mathrm{1}}{}^{}+I{}_{\mathrm{2}}{}^{}+I{}_{\mathrm{3}}{}^{}+I{}_{\mathrm{4}}{}^{}+I_{\mathrm{5}}.\end{array}$(12)

For $v>\gamma ,\mathrm{0}<\gamma <\mathrm{1},$we have

$\begin{array}{l}{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}{X}_j|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }={\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}{\displaystyle \sum _{l=\mathrm{1}}^{\mathrm{5}}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)})|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }\\ \le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{{\displaystyle \sum _{l=\mathrm{1}}^{\mathrm{5}}}\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)})|-\epsilon n^{\alpha }{\}}_{+}^{\gamma }\\ \le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})})|+{\displaystyle \sum _{l=\mathrm{2}}^{\mathrm{5}}}{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}|a_i||{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}{Y}_j^{(n,l)}|-\frac{\mathrm{3}}{\mathrm{4}}\epsilon n^{\alpha }{\}}_{+}^{\gamma }\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})}{)|\}}^v\\ +C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}E\{{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}(Y_j^{(n,l)}-E{Y}_j^{(n,l)}{)|\}}^v{+C{\displaystyle \sum _{l=\mathrm{4}}^{\mathrm{5}}}{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha \gamma -\mathrm{2}}E\{{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i||{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}{Y}_j^{(n,l)}|\}}^{\gamma }\\ =:I{}_{\mathrm{1}}{}^{}+I{}_{\mathrm{2}}{}^{}+I{}_{\mathrm{3}}{}^{}+I{}_{\mathrm{4}}{}^{}+I_{\mathrm{5}}.\end{array}$(13)

In order to prove (6) of Theorem 1, we need to prove $I_i<\mathrm{\infty },i=\mathrm{1,2},\cdots ,\mathrm{5}.$

For $I{}_{\mathrm{1}}{}^{},$by Lemma 5, Markov and $\mathrm{H}\ddot{\mathrm{o}}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}$ inequalities, we have that for any $v>\mathrm{m}\mathrm{a}\mathrm{x}\{\mathrm{2},\gamma ,p\}$,

$\begin{array}{l}{I}_{\mathrm{1}}=C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}E\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}{a}_i{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})}{)|\}}^v\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}E\{{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})}{)|\}}^v\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}E\{{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i{|}^{\mathrm{1}-\mathrm{1}/v}(|a_i{|}^{\mathrm{1}/v}\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})}{)|\}}^v\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}({\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}|a_i{|)}^{v-\mathrm{1}}({\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}}|a_i|E\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=i+\mathrm{1}}^{i+k}}(Y_j^{(n,\mathrm{1})}-E{Y}_j^{(n,\mathrm{1})}){|}^v)\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|lo{g}^{v}n\{{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}E|Y_j^{(n,\mathrm{1})}{|}^v+g(n)({\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}E|Y_j^{(n,\mathrm{1})}{|}^{\mathrm{2}}{)}^{v/\mathrm{2}}\}\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}\mathrm{l}\mathrm{o}{\mathrm{g}}^{v}n{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|\{{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}[E|Y_j{|}^{v}I(|Y_j|\le n^{\alpha q})+n^{v\alpha q}I(|Y_j|>n^{\alpha q})]\}\\ +C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}}g(n)\mathrm{l}\mathrm{o}{\mathrm{g}}^{v}n{\displaystyle \sum _{i=-\infty }^{\infty }}|a_i|\{{\displaystyle \sum _{j=i+\mathrm{1}}^{i+n}}[E|Y_j{|}^{\mathrm{2}}I(|Y_j|\le n^{\alpha q})+n^{\mathrm{2}\alpha q}I(|Y_j|>n^{\alpha q}{)]\}}^{v/\mathrm{2}}\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{1}}\mathrm{l}\mathrm{o}{\mathrm{g}}^{v}n[E|Y{|}^{v}I(|Y|\le n^{\alpha q})+n^{v\alpha q}I(|Y|>n^{\alpha q})]+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\alpha v-\mathrm{2}+v/\mathrm{2}}g(n)\mathrm{l}\mathrm{o}{\mathrm{g}}^{v}n[E|Y{|}^{\mathrm{2}}I(|Y|\le n^{\alpha q})+n^{\mathrm{2}\alpha q}I(|Y|>n^{\alpha q}{)]}^{v/\mathrm{2}}\\ =:I_{\mathrm{11}}+I_{\mathrm{12}}.\end{array}$(14)