© Wuhan University 2025

0 Introduction

In many probabilistic and statistical models, random variables are dependent. Consequently, scholars have introduced many types of dependent random variables, such as negatively associated (NA) random variables, negatively orthant dependent (NOD) random variables, and extend negatively dependent (END) random variables and so on. Among these, widely orthant dependent (WOD) random variables represent one of the most general forms of dependence. They were first introduced by Wang et al^[1], defined as follows:

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 exist finite sequences of real numbers $\left\{g_{\mathrm{U}}\left(n\right),n\ge \mathrm{1}\right\}$ such that for each $n\ge \mathrm{1}$, $x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_n\in \mathbb{R}$,

$P\left(X_{\mathrm{1}}>x_{\mathrm{1}},X_{\mathrm{2}}>x_{\mathrm{2}},\dots ,X_n>x_n\right)\le g_{\mathrm{U}}\left(n\right){\displaystyle \prod _{i=\mathrm{1}}^n}P\left(X_i>x_i\right).$

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

$P\left(X_{\mathrm{1}}\le x_{\mathrm{1}},X_{\mathrm{2}}\le x_{\mathrm{2}},\dots ,X_n\le x_n\right)\le g_{\mathrm{L}}\left(n\right){\displaystyle \prod _{i=\mathrm{1}}^n}P\left(X_i>x_i\right).$

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

When $g_{\mathrm{U}}\left(n\right)=g_{\mathrm{L}}\left(n\right)=\mathrm{1}$, $\left\{X_n,n\ge \mathrm{1}\right\}$ are NOD random variables^[2]. When $g_{\mathrm{U}}\left(n\right)=g_{\mathrm{L}}\left(n\right)=M\ge \mathrm{1}$, $\left\{X_n,n\ge \mathrm{1}\right\}$ are END random variables^[3]. Therefore, WOD random variables represent a broad structure of dependent random variables.

Since the concept of WOD random variables was introduced, many scholars have devoted efforts to studying their limit properties and applications, achieving significant results. For example, Wang et al^[4] obtained the precise large deviations; Qiu et al^[5] established the complete convergence and moment complete convergence of the weighted sums; Liu et al^[6] derived the moment complete convergence; Wang et al^[7] and Chen et al^[8] studied the asymptotic of ruin probabilities in renewal risk models based on WOD sequences; Shen^[9] proved the Bernstein-type probability inequality; Wang et al^[10] investigated complete convergence and its applications in nonparametric regression models; Ding et al^[11] provided results on the complete convergence of weighted sums; Song et al^[12-14] analyzed the convergence of moving average processes generated by WOD random variables, and so on.

Inspired by m-END and WOD dependence structures, Fang et al^[15] introduced the concept of m-WOD random variables, defined as follows:

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

From the definition, we see that m-WOD random variables represent a broader class of dependence than WOD random variables. Therefore, investigating the complete convergence of m-WOD random variables is very interesting.

It is well-known that for sequencse of independent and identically distributed random variables $\left\{X_n,X,n\ge \mathrm{1}\right\}$, Spitzer^[16] proved that $EX=\mathrm{0}$ is equivalent to

${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{-\mathrm{1}}P\left(|{\displaystyle \sum _{j=\mathrm{1}}^n}{X}_j|>\epsilon n\right)<\mathrm{\infty },\forall \epsilon >\mathrm{0}.$(1)

and (1) is equivalent to

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

However, the converse does not hold for the dependent case, as shown in Ref. [17]. Therefore, it is more interesting to investigate (2) than (1), and we note that (2) implies Kolmogorov’s strong law of large numbers

$\frac{\mathrm{1}}{n}{\displaystyle \sum _{j=\mathrm{1}}^n}{X}_j\to \mathrm{0},\mathrm{a}.\mathrm{s}.$

Recently, Chen et al^[18-19] obtained Spitzer’s law for maximum partial sums of WOD random variables. Inspired by the above study, this paper aims to generalize Chen’s results^[18-19] to cases of complete convergence for m-WOD random variables.

Definition 3   The random variables $\left\{X_n,n\ge \mathrm{1}\right\}$ are called to be stochastically dominated by a random variable $X$, if for any $x>\mathrm{0}$,

$P\left(\left|X_n\right|>x\right)\le CP\left(\left|X\right|>x\right),n\ge \mathrm{1}$

where the constant $C>\mathrm{0}$.

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

1 Some Lemmas and Main Results

Lemma 1^[15] The sequences $\left\{X_n,n\ge \mathrm{1}\right\}$ are m-WOD random variables, if the functions $\left\{f_n,n\ge \mathrm{1}\right\}$ are non-decreasing (non-increasing), then $\left\{f_n\left(X_n\right),n\ge \mathrm{1}\right\}$ are also m-WOD random variables sequences with same dominating coefficients.

Lemma 2^[15] The sequences $\left\{X_n,n\ge \mathrm{1}\right\}$ are m-WOD random variables with dominating coefficients $g\left(n\right)$. For every $j\ge \mathrm{1}$, the $E{X}_j=\mathrm{0}$ and $E|X_j{|}^p<\mathrm{\infty }$. Then, there exist positive constants $C_{\mathrm{1}}=C_{\mathrm{1}}\left(p,m\right),C_{\mathrm{2}}=C_{\mathrm{2}}\left(p,m\right),$ depending only on $p$ and $m$, such that

$E\left({\displaystyle \sum _{j=\mathrm{1}}^n}|X_j{|}^p\right)\le \left[C_{\mathrm{1}}\left(p,m\right)+C_{\mathrm{2}}\left(p,m\right)g\left(n\right)\right]{\displaystyle \sum _{j=\mathrm{1}}^n}E|X_j{|}^p,\mathrm{1}<p\le \mathrm{2};$

$E\left({\displaystyle \sum _{j=\mathrm{1}}^n}|X_j{|}^p\right)\le C_{\mathrm{1}}\left(p,m\right){\displaystyle \sum _{j=\mathrm{1}}^n}E|X_j{|}^p+C_{\mathrm{2}}\left(p,m\right)g\left(n\right){\left({\displaystyle \sum _{j=\mathrm{1}}^n}E{X}_j^{\mathrm{2}}\right)}^{p/\mathrm{2}},p>\mathrm{2}.$

Lemma 3^[15] The sequences $\left\{X_n,n\ge \mathrm{1}\right\}$ are m-WOD random variables with dominating coefficients $g\left(n\right)$. For every $j\ge \mathrm{1}$, the $E{X}_j=\mathrm{0}$ and $E|X_j{|}^p<\mathrm{\infty }$. Then, there exist positive constants $C_{\mathrm{1}}=C_{\mathrm{1}}\left(p,m\right),C_{\mathrm{2}}=C_{\mathrm{2}}\left(p,m\right),$ depending only on $p$ and $m$, such that

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

$E\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}{X}_j{|}^p\right)\le C_{\mathrm{1}}\left(p,m\right){\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^p{\displaystyle \sum _{j=\mathrm{1}}^n}E|X_j{|}^p+C_{\mathrm{2}}\left(p,m\right)g\left(n\right){\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^p{\left({\displaystyle \sum _{j=\mathrm{1}}^n}E{X}_j^{\mathrm{2}}\right)}^{p/\mathrm{2}},p>\mathrm{2}.$

Lemma 4^[20] Constant $a>\mathrm{0},b>\mathrm{0}$, let $\left\{X_n,n\ge \mathrm{1}\right\}$ be stochastically dominated by X, then there exist positive constants $C_{\mathrm{1}},C_{\mathrm{2}}$ such that the following inequalities are established:

$E|X_n{|}^{a}I\left(|X_n|\le b\right)\le C_{\mathrm{1}}\left[E|X{|}^{a}I\left(|X|\le b\right)+b^{a}I\left(|X|>b\right)\right],$

$E|X_n{|}^{a}I\left(|X_n|>b\right)\le C_{\mathrm{2}}E|X{|}^{a}I\left(|X|>b\right).$

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

Theorem 1   Let $\left\{X_n,n\ge \mathrm{1}\right\}$ be sequences of m-WOD random variables stochastically dominated by a random variable X with dominating coefficients $g\left(n\right).$ Let $\mathrm{0}<\mathrm{1}/p\le \alpha <\mathrm{1},$ $E|X{|}^p<\mathrm{\infty }.$ Assume that one of the following conditions holds:

${\mathrm{A}}_{\mathrm{1}}:$ Let $g(x)$ be a nondecreasing positive function on $[\mathrm{0},+\mathrm{\infty })$, such that $g(x)/x^{\tau }\downarrow \mathrm{0}$ for some $\tau >\mathrm{0}$.

${\mathrm{A}}_{\mathrm{2}}:$ Let $h(x)$ be a nondecreasing positive function on $[\mathrm{0},+\mathrm{\infty })$, such that $h(x)/x\downarrow \mathrm{0}$ and ${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}\frac{g(n)}{n{h}^{\gamma }(n)}<\mathrm{\infty }$ for some $\gamma >\mathrm{0}$.

Then, for $\theta >\mathrm{m}\mathrm{a}\mathrm{x}\{\mathrm{1}/\mathrm{2},\alpha \},$

${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(X_j-E{X}_j)|>\epsilon n^{\theta }\right)<\mathrm{\infty },\forall \epsilon >\mathrm{0}.$(3)

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

Theorem 2   Let $\left\{X_n,n\ge \mathrm{1}\right\}$ be sequences of m-WOD random variables stochastically dominated by a random variable X with dominating coefficients $g(n),$ $E|X{|}^{p+\delta }<\mathrm{\infty },$ $p>\mathrm{1},$ $\mathrm{0}<\delta <p(p-\mathrm{1}).$ Assume ${\mathrm{A}}_{\mathrm{1}}$ or ${\mathrm{A}}_{\mathrm{2}}$ holds. Then, for $\theta >\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{1}/\mathrm{2},p/(p+\delta )\right\},$

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

Theorem 3   Let $\left\{X_n,n\ge \mathrm{1}\right\}$ be sequences of m-WOD random variables stochastically dominated by a random variable X with dominating coefficients $g(n),$ $E|X{|}^p<\mathrm{\infty },$ $p>\mathrm{1}.$ Assume ${\mathrm{A}}_{\mathrm{1}}$ or ${\mathrm{A}}_{\mathrm{2}}$ holds. Then

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

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

Theorem 4   Let $\left\{X_n,n\ge \mathrm{1}\right\}$ be sequences of m-WOD random variables stochastically dominated by a random variable X with dominating coefficients $g(n),$ $E|X|h^{\delta }(|X|)<\mathrm{\infty },\mathrm{0}<\delta \le \mathrm{1}.$ Assume ${\mathrm{A}}_{\mathrm{2}}$ holds and

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{h(x)}{h(x/h(x))}<\mathrm{\infty },$(6)

then

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

Remark 1   The m-WOD random variables encompass WOD, m-NA, m-NOD, m-END, among others. Thus, the results in this paper extend and improve upon existing results.

Remark 2   Since stochastic domination is a weaker condition than identical distribution, the results in this paper also hold under the condition of identical distribution.

Remark 3   Taking $\delta =\mathrm{1}$ in Theorem 4, we obtain the result of Theorem 1.1 in Ref. [17]. Taking $\delta =\mathrm{1}/\mathrm{2},h(x)=g(x)$ in Theorem 4, we obtain the result of Theorem 1.2 in Chen et al’s ^[18]. Therefore, our results extend and improve the results in Refs. [17-18].

2 Proof of Theorems

Proof of Theorem 1   Noting $X_i=X_i^{+}-X_i^{-}$, hence to prove (3), it only to prove that, for any $\epsilon >\mathrm{0}$,

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

and

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

By Lemma 1, $\left\{X_n^{+},n\ge \mathrm{1}\right\}$ and $\left\{X_n^{-},n\ge \mathrm{1}\right\}$ are also m-WOD random variables sequences with dominating coefficients $g(n),n\ge \mathrm{1}.$ Therefore, without loss of generality, we assume $X_n\ge \mathrm{0},n\ge \mathrm{1}$.

For a fixed $n\ge \mathrm{1}$, for $\mathrm{1}\le j\le n$, denote

$Y_{nj}=X_{j}I\left(X_j\le n^{\alpha }\right)+n^{\alpha }I\left(X_j>n^{\alpha }\right),Z_{nj}=X_j-Y_{nj}=\left(X_j-n^{\alpha }\right)I\left(X_j>n^{\alpha }\right).$

Then

$\begin{array}{l}{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(X_j-E{X}_j)|>\epsilon n^{\theta }\right)\le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(Y_{nj}-E{Y}_{nj})|>\epsilon n^{\theta }/\mathrm{2}\right)+{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}}P\left({\displaystyle \underset{j=\mathrm{1}}{\overset{n}{\cup }}}\{X_j>n^{\alpha }\}\right)\\ \le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(Y_{nj}-E{Y}_{nj})|>\epsilon n^{\theta }/\mathrm{2}\right)+{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}}{\displaystyle \sum _{j=\mathrm{1}}^n}P\left(X_j>n^{\alpha }\right)=:I_{\mathrm{1}}+I_{\mathrm{2}}.\end{array}$

Thus, to prove (3), we only need to show $I_{\mathrm{1}}<\mathrm{\infty }$ and $I_{\mathrm{2}}<\mathrm{\infty }$.

Combining with Lemma 4 and condition $\alpha p-\mathrm{1}>-\mathrm{1}$, we obtain

$I_{\mathrm{2}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{1}}P\left(X>n^{\alpha }\right)=C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{1}}{\displaystyle \sum _{k=n}^{\mathrm{\infty }}}P\left(k^{\alpha }<X\le {(k+\mathrm{1})}^{\alpha }\right)$

$=C{\displaystyle \sum _{k=\mathrm{1}}^{\infty }}P\left(k^{\alpha }<X\le {(k+\mathrm{1})}^{\alpha }\right){\displaystyle \sum _{n=\mathrm{1}}^k}{n}^{\alpha p-\mathrm{1}}\le C{\displaystyle \sum _{k=\mathrm{1}}^{\infty }}{k}^{\alpha p}P\left(k^{\alpha }<X\le {(k+\mathrm{1})}^{\alpha }\right)\le CE|X{|}^p<\mathrm{\infty }.$(8)

By Lemma 1, $\left\{Y_{nj}-E{Y}_{nj}\right\}$ are also m-WOD random variables with the same dominating coefficients.

For $I_{\mathrm{1}}$, by Markov’s inequality and Lemma 3 , we have that for any $v>\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{2},p\right\}$,

$I_{\mathrm{1}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v}E\left\{\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(Y_{nj}-E{Y}_{nj}){|}^v\right\}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v\left\{{\displaystyle \sum _{j=\mathrm{1}}^n}E|Y_{nj}{|}^v+g\left(n\right){\left({\displaystyle \sum _{j=\mathrm{1}}^n}E|Y_{nj}{|}^{\mathrm{2}}\right)}^{v/\mathrm{2}}\right\}=:I_{\mathrm{11}}+I_{\mathrm{12}}.$(9)

For $I_{\mathrm{11}}$, combining with Lemma 4 and the condition $\theta >\left\{\mathrm{1}/\mathrm{2},\alpha \right\},$ we obtain

$I_{\mathrm{11}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v{\displaystyle \sum _{j=\mathrm{1}}^n}\left[E{X}_j^{v}I\left(X_j\le n^{\alpha }\right)+n^{v\alpha }P\left(X_j>n^{\alpha }\right)\right]\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{1}-\theta v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v\left[E{X}^{v}I\left(X\le n^{\alpha }\right)+n^{v\alpha }P\left(X>n^{\alpha }\right)\right]$

$\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{1}-\theta v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v\left[E{X}^p{n}^{\alpha (v-p)}I\left(X\le n^{\alpha }\right)+E{X}^p{n}^{\alpha (v-p)}P\left(X>n^{\alpha }\right)\right]\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{-\mathrm{1}-(\theta -\alpha )v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v<\mathrm{\infty }.$(10)

For $I_{\mathrm{12}}$, we have

$I_{\mathrm{12}}\le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g(n){｛{\displaystyle \sum _{j=\mathrm{1}}^n}\left[E{X_j}^{\mathrm{2}}I\left(X_j\le n^{\alpha }\right)+n^{\mathrm{2}\alpha }P\left(X_j>n^{\alpha }\right)\right]｝}^{v/\mathrm{2}}$

$\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g(n){\left[E{X}^{\mathrm{2}}I\left(X\le n^{\alpha }\right)+n^{\mathrm{2}\alpha }P\left(X>n^{\alpha }\right)\right]}^{v/\mathrm{2}}.$(11)

To prove $I_{\mathrm{12}}<\mathrm{\infty }$, we consider two cases:

Case 1: When $p\ge \mathrm{2}$, the condition $E|X{|}^p<\mathrm{\infty }$implies $E|X{|}^{\mathrm{2}}<\mathrm{\infty }$, taking $v>\mathrm{m}\mathrm{a}\mathrm{x}\left\{p,\frac{\mathrm{2}\left(\alpha p-\mathrm{1}+\tau \right)}{\mathrm{2}\theta -\mathrm{1}},\frac{\mathrm{2}\left(\alpha p+\gamma \right)}{\mathrm{2}\theta -\mathrm{1}}\right\}$, then

$I_{\mathrm{12}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g(n){\left[E{X}^{\mathrm{2}}\right]}^{v/\mathrm{2}}\le \{\begin{array}{l}C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}+\tau }{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v,\mathrm{i}\mathrm{f}{\mathrm{A}}_{\mathrm{1}}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\\ C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}+\mathrm{1}+\gamma }{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v\frac{g(n)}{n{h}^{\gamma }(n)},\mathrm{i}\mathrm{f}{\mathrm{A}}_{\mathrm{2}}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\end{array}<\mathrm{\infty }.$(12)

Case 2: When $\mathrm{1}<p<\mathrm{2}$, under $E|X{|}^p<\mathrm{\infty }$, taking $v>\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{2},\frac{\mathrm{2}\left(\alpha p-\mathrm{1}+\tau \right)}{\mathrm{2}\left(\theta -\alpha \right)+\left(\alpha p-\mathrm{1}\right)},\frac{\mathrm{2}\left(\alpha p+\gamma \right)}{\mathrm{2}\left(\theta -\alpha \right)+\left(\alpha p-\mathrm{1}\right)}\right\}$, we obtain

$\begin{array}{l}{I}_{\mathrm{12}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g(n){\left[E{X}^p{n}^{\alpha (\mathrm{2}-p)}I\left(X\le n^{\alpha }\right)+E{X}^p{n}^{\alpha (\mathrm{2}-p)}P\left(X>n^{\alpha }\right)\right]}^{v/\mathrm{2}}\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g(n)n^{\alpha (\mathrm{2}-p)v/\mathrm{2}}{\left(E{X}^p\right)}^{v/\mathrm{2}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}+\alpha v-\alpha pv/\mathrm{2}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g\left(n\right)\end{array}$

$\le \{\begin{array}{l}C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}+\alpha v-\alpha pv/\mathrm{2}+\tau }{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v,\mathrm{i}\mathrm{f}{\mathrm{A}}_{\mathrm{1}}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\\ C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{\alpha p-\mathrm{2}-\theta v+v/\mathrm{2}+\alpha v-\alpha pv/\mathrm{2}+\mathrm{1}+\gamma }{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v\frac{g(n)}{n{h}^{\gamma }(n)},\mathrm{i}\mathrm{f}{\mathrm{A}}_{\mathrm{2}}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\end{array}<\mathrm{\infty }.$(13)

From (8)-(13), the proof of Theorem 1 is completed.

Proof of Theorem 2   For fixed $n\ge \mathrm{1}$, $\mathrm{1}\le j\le n,$ denote

$\begin{array}{l}{Y}_{nj}=X_{j}I\left(X_j\le n^{p/\left(p+\delta \right)}\right)+n^{p/\left(p+\delta \right)}I\left(X_j>n^{p/\left(p+\delta \right)}\right),\\ Z_{nj}=X_j-Y_{nj}=\left(X_j-n^{p/\left(p+\delta \right)}\right)I\left(X_j>n^{p/\left(p+\delta \right)}\right).\end{array}$

The method and the proof for Theorem 2 are the same as Theorem 1, so they are omitted.

Proof of Theorem 3   The condition $E|X{|}^p<\mathrm{\infty },p>\mathrm{1},$ implies that $E|X|<\mathrm{\infty }$. Consequently, there exists a positive integer $N$ such that $EXI(X>N)<\epsilon /\mathrm{8}$.

For $j\ge \mathrm{1}$, let

$Y_j=X_{j}I\left(X_j\le N\right)+NI\left(X_j>N\right),Z_j=X_j-Y_j=\left(X_j-N\right)I\left(X_j>N\right).$

Then

${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(X_j-E{X}_j)|>\epsilon n\right)\le {\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(Y_j-E{Y}_j)|>\epsilon n/\mathrm{2}\right)+{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(Z_j-E{Z}_j)|>\epsilon n/\mathrm{2}\right)$

$=:I_{\mathrm{3}}+I_{\mathrm{4}}.$(14)

Thus, to prove (14), we only need to show that $I_{\mathrm{3}}<\mathrm{\infty }$ and $I_{\mathrm{4}}<\mathrm{\infty }$.

For $I_{\mathrm{3}}$, noting $\left\{(Y_j-E{Y}_j),j\ge \mathrm{1}\right\}$ are sequences of m-WOD with dominating coefficient $g(n)$ for $n\ge \mathrm{1}$ by Lemma 1. By Markov’s inequality, Lemma 3 and Lemma 4, taking $v>\mathrm{m}\mathrm{a}\mathrm{x}\left\{p,\mathrm{2},\mathrm{2}(p-\mathrm{1}+\tau ),\mathrm{2}(p+\gamma )\right\}$, we have

$I_{\mathrm{3}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}-v}E｛\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \sum _{j=\mathrm{1}}^k}(Y_j-E{Y}_j){|}^v｝\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}-v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v｛{\displaystyle \sum _{j=\mathrm{1}}^n}E|Y_j{|}^v+g\left(n\right){\left({\displaystyle \sum _{j=\mathrm{1}}^n}|E{Y}_j{|}^{\mathrm{2}}\right)}^{v/\mathrm{2}}｝\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}-v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v{\displaystyle \sum _{j=\mathrm{1}}^n}\left[E{X_j}^{v}I\left(X_j\le N\right)+N^{v}P\left(X_j>N\right)\right]$

$\begin{array}{l}+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}-v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g\left(n\right){\left[{\displaystyle \sum _{j=\mathrm{1}}^n}E{X_j}^{\mathrm{2}}I\left(X_j\le N\right)+N^{\mathrm{2}}P\left(X_j>N\right)\right]}^{v/\mathrm{2}}\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{1}-v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}-v/\mathrm{2}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^{v}g(n)\end{array}$

$\le \{\begin{array}{l}C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{1}-v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}-v/\mathrm{2}+\tau }{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v,\mathrm{i}\mathrm{f}{\mathrm{A}}_{\mathrm{1}}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\\ C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{1}-v}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}-v/\mathrm{2}+\gamma +\mathrm{1}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}n\right)}^v\frac{g(n)}{n{h}^{\gamma }(n)},\mathrm{i}\mathrm{f}{\mathrm{A}}_{\mathrm{2}}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\end{array}<\mathrm{\infty }.$(15)

For $n>N,j\ge \mathrm{1},$ let

$Z_{nj}^{\text{'}}=\left(X_j-N\right)I\left(N<X_j\le n\right)+\left(n-N\right)I\left(X_j>n\right).$

By definition of $Z_j$ and $Z_{nj}^{\text{'}}$, we get

${\displaystyle \sum _{j=\mathrm{1}}^n}E{Z}_j\le {\displaystyle \sum _{j=\mathrm{1}}^n}E{X}_{j}I\left(X_j>N\right)<n\epsilon /\mathrm{8},{\displaystyle \sum _{j=\mathrm{1}}^n}E{Z}_{nj}^{\text{'}}\le {\displaystyle \sum _{j=\mathrm{1}}^n}E{X}_{j}I\left(X_j>N\right)<n\epsilon /\mathrm{8}.$

For $I_{\mathrm{4}}$, we obtain

$\begin{array}{l}{I}_{\mathrm{4}}\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left({\displaystyle \sum _{j=\mathrm{1}}^n}{Z}_j+{\displaystyle \sum _{j=\mathrm{1}}^n}E{Z}_j>n\epsilon /\mathrm{2}\right)\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left({\displaystyle \sum _{j=\mathrm{1}}^n}{Z}_j>\mathrm{3}n\epsilon /\mathrm{8}\right)\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left({\displaystyle \sum _{j=\mathrm{1}}^n}{Z}_{nj}^{\text{'}}>\mathrm{3}n\epsilon /\mathrm{8}\right)+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left({\displaystyle \underset{j=\mathrm{1}}{\overset{n}{\cup }}}{X}_j>n\right)\\ \le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left(|{\displaystyle \sum _{j=\mathrm{1}}^n}(Z_{nj}^{\text{'}}-E{Z}_{nj}^{\text{'}})|+{\displaystyle \sum _{j=\mathrm{1}}^n}E{Z}_{nj}^{\text{'}}>\mathrm{3}n\epsilon /\mathrm{8}\right)+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{1}}P\left(X>n\right)\end{array}$

$\le C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{2}}P\left(|{\displaystyle \sum _{j=\mathrm{1}}^n}(Z_{nj}^{\text{'}}-E{Z}_{nj}^{\text{'}})|>n\epsilon /\mathrm{4}\right)+C{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{n}^{p-\mathrm{1}}P\left(X>n\right)=:I_{\mathrm{41}}+I_{\mathrm{42}}.$(16)