On the complete convergence for arrays of rowwise ψ-mixing random variables

Abstract

Some sufficient conditions for complete convergence for maximal weighted sums ${max}_{1\le j\le n}|{\sum }_{k=1}^j{a}_{nk}{X}_{nk}|$ and weighted sums ${\sum }_{k=1}^n{a}_{nk}{X}_{nk}$ are presented, where $\{X_{nk},1\le k\le n,n\ge 1\}$ is an array of rowwise ψ-mixing random variables, and $\{a_{nk},1\le k\le n,n\ge 1\}$ is an array of constants. The obtained results extend and improve the corresponding result in the previous literature.

MSC:60F15.

1 Introduction

The following notion was given firstly by Hsu and Robbins [1].

Definition 1.1 A sequence of random variables $\{U_n,n\ge 1\}$ is said to converge completely to a constant θ if for any $\epsilon >0$,

$$\sum _{n=1}^{\mathrm{\infty }}P(|U_n-\theta |>\epsilon )<\mathrm{\infty }.$$

In this case, we write $U_n\to \theta$ completely. In view of the Borel-Cantelli lemma, the result above implies that $U_n\to \theta$ almost surely. Therefore, the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables as well as weighted sums of random variables.

Let $\{X_n,n\ge 1\}$ be a sequence of random variables, defined on a probability space $(\mathrm{\Omega },\mathcal{F},P)$, and denote σ-algebras

$${\mathcal{F}}_n^m=\sigma (X_k,n\le k\le m),1\le n\le m\le \mathrm{\infty }.$$

As usual, for a σ-algebra ℱ, we denote by ${\mathcal{L}}^2(\mathcal{F})$ the class of all ℱ-measurable random variables with the finite second moment. Given σ-algebras , ℬ in ℱ, let

$$\begin{array}{c}\psi (\mathcal{A},\mathcal{B})=\underset{A\in \mathcal{A},B\in \mathcal{B},P(A)P(B)>0}{sup}|\frac{P(AB)}{P(A)P(B)}-1|,\hfill \\ \phi (\mathcal{A},\mathcal{B})=\underset{A\in \mathcal{A},B\in \mathcal{B},P(A)>0}{sup}|P(B|A)-P(B)|.\hfill \end{array}$$

Define the mixing coefficients by

$$\psi (n)=\underset{k\ge 1}{sup}\psi ({\mathcal{F}}_1^k,{\mathcal{F}}_{k+n}^{\mathrm{\infty }}),\phi (n)=\underset{k\ge 1}{sup}\phi ({\mathcal{F}}_1^k,{\mathcal{F}}_{k+n}^{\mathrm{\infty }}),n\ge 0.$$

The concepts of ψ-mixing and φ-mixing random variables were introduced by Blum et al. [2] and Dobrushin [3], respectively.

Definition 1.2 A sequence of random variables $\{X_n,n\ge 1\}$ is said to be a ψ-mixing (φ-mixing) sequence of random variables if $\psi (n)\downarrow 0$ ($\phi (n)\downarrow 0$) as $n\to \mathrm{\infty }$.

Clearly, from the definition above, we know that the independence implies ψ-mixture and φ-mixture. It is easily seen that the ψ-mixing condition is stronger than the φ-mixing. Therefore, the family of ψ-mixing is a special case of φ-mixing. Years after the appearance of Dobrushin [3], many works of investigation concerning the convergence properties of φ-mixing random variables have emerged. We refer the reader to Ibragimov [4], Cogburn [5], Sen [6], Choi and Sung [7], Utev [8], Chen [9], Shao [10], Rüdiger [11], Chen et al. [12], Zhou [13], Wang et al. [14, 15], Guo [16].

However, according to our knowledge, few papers discuss the subjects for sequences or arrays of ψ-mixing random variables except Blum et al. [2], Bradley [17], Yang [18], Wu and Zhu [19], Wang et al. [14, 15], and Yang and Liu [20]. The goal of this paper is to study a complete convergence for arrays of rowwise ψ-mixing random variables.

Then we recall that the following concept of stochastic domination is a slight generalization of identical distribution.

Definition 1.3 An array of rowwise random variables $\{X_{nk},1\le k\le n,n\ge 1\}$ is said to be stochastically dominated by a nonnegative random variable X (write $\{X_{nk}\}\prec X$) if there exists a constant $C>0$ such that

$$\underset{n,k}{sup}P(|X_{nk}|>x)\le CP(X>x),\mathrm{\forall }x>0.$$

Stochastic dominance of $\{X_{nk},1\le k\le n,n\ge 1\}$ by the random variable X implies that $E{|X_{nk}|}^p\le CE{X}^p$ if the p-moment of X exists, i.e., if $E{X}^p<\mathrm{\infty }$.

Hu et al. [21] obtained the following result in the complete convergence.

Theorem A Let $\{X_{nk},1\le k\le n,n\ge 1\}$ be an array of rowwise independent random variables with $E{X}_{nk}=0$. Suppose that $\{X_{nk},1\le k\le n,n\ge 1\}$ are uniformly bounded by some random variable X. If $E{|X|}^{2p}<\mathrm{\infty }$ for some $1\le p<2$, then

$$n^{-1/p}\sum _{k=1}^n{X}_{nk}\to 0\mathit{\text{completely}}.$$

Taylor et al. [22], Baek et al. [23] extended and generalized Theorem A to rowwise negatively dependent (ND) random variables.

The main purpose of this article is to discuss the complete convergence for weighted sums of ψ-mixing random variables. We shall extend Theorem A by considering ψ-mixing instead of independent. It is worthy to point out that our main methods differ from those used by Hu et al. [21].

Below, C will be used to denote various positive constants, whose value may vary from one application to another. For a finite set A, the symbol $\mathrm{♯}(A)$ denotes the number of elements in the set A. $I_{(A)}$ will indicate the indicator function of A.

2 Main results and some lemmas

Now, we state our main results. The proofs will be given in Section 3.

Theorem 2.1 Let $\{X_{nk},1\le k\le n,n\ge 1\}$ be an array of rowwise ψ-mixing random variables with ${\sum }_{m=1}^{\mathrm{\infty }}\psi (m)<\mathrm{\infty }$ and $E{X}_{nk}=0$. Suppose that $\{X_{nk}\}\prec X$ and $E{X}^{2p}<\mathrm{\infty }$ for some $p>0$. Let $\{a_{nk},1\le k\le n,n\ge 1\}$ be a real numbers array satisfying ${max}_{1\le k\le n}|a_{nk}|=O(n^{-\alpha })$ for some $\alpha >1/(2p)$. Furthermore, when $p\ge 1$, we suppose that there exists a constant $\theta >0$ such that ${\sum }_{k=1}^n{a}_{nk}^2\le C{n}^{-\theta }$. Then

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

Take $a_{nk}=n^{-1/p}$ and $1\le p<2$ in Theorem 2.1, we can have the following corollary.

Corollary 2.1 Let $\{X_{nk},1\le k\le n,n\ge 1\}$ be an array of rowwise ψ-mixing random variables with ${\sum }_{m=1}^{\mathrm{\infty }}\psi (m)<\mathrm{\infty }$ and $E{X}_{nk}=0$. Suppose that $\{X_{nk}\}\prec X$ and $E{X}^{2p}<\mathrm{\infty }$ for some $1\le p<2$, then

$$\sum _{n=1}^{\mathrm{\infty }}P(\underset{1\le j\le n}{max}\left|\sum _{k=1}^j{X}_{nk}\right|>n^{1/p}\epsilon )<\mathrm{\infty },\mathrm{\forall }\epsilon >0.$$

Remark 2.1 Since the independence implies ψ-mixture, Theorem 2.1 and Corollary 2.1 hold for arrays of rowwise independent random variables. Therefore, Theorem 2.1 and Corollary 2.1 extend and improve Theorem A.

Theorem 2.2 Let $\{X_{nk},1\le k\le n,n\ge 1\}$ be an array of rowwise ψ-mixing random variables with $E{X}_{nk}=0$. Suppose that $\{X_{nk}\}\prec X$ and $E{X}^{2p}<\mathrm{\infty }$ for some $p\ge 1$. Let $\{a_{nk},1\le k\le n,n\ge 1\}$ be a real numbers array satisfying ${max}_{1\le k\le n}|a_{nk}|=O(n^{-\alpha })$ for some $\alpha >1/(2p)$. Suppose that the following statements hold.

• (i) There exists a positive constant $\lambda <min\{\frac{1}{2p},\frac{2\alpha p-1}{4p}\}$ such that ${\sum }_{n=1}^{\mathrm{\infty }}{\psi }^{\frac{\lambda }{1-\lambda }}(n)<\mathrm{\infty }$;

• (ii) $logn{\sum }_{k=1}^n{a}_{nk}^2=o(1)$ if $\frac{1}{2p}<\alpha \le \frac{1}{2}$. Then

  $$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\left|\sum _{k=1}^n{a}_{nk}{X}_{nk}\right|>\epsilon )<\mathrm{\infty },\mathrm{\forall }\epsilon >0.$$

Remark 2.2 Compared with Theorem 2.1, Theorem 2.2 requires a stronger mixing rate, but weakens the requirement of ${\sum }_{k=1}^n{a}_{nk}^2$. In fact, $logn{\sum }_{k=1}^n{a}_{nk}^2=o(1)$ holds if ${\sum }_{k=1}^n{a}_{nk}^2\le C{n}^{-\theta }$, $\theta >0$.

Now, we state some lemmas which will be used in the proofs of our main results.

Lemma 2.1 (Wang et al. [14, 15])

Let $\{X_n,n\ge 1\}$ be a sequence of ψ-mixing random variables satisfying ${\sum }_{m=1}^{\mathrm{\infty }}\psi (m)<\mathrm{\infty }$, $q\ge 2$. Assume that $E{X}_n=0$ and $E{|X_n|}^q<\mathrm{\infty }$ for each $n\ge 1$. Then there exists a constant C depending only on q and $\psi (\cdot )$ such that

$$E\underset{1\le j\le n}{max}{\left|\sum _{k=1}^j{X}_k\right|}^q\le C\{\sum _{k=1}^{n}E{|X_k|}^q+{\left(\sum _{k=1}^{n}E{X}_k^2\right)}^{q/2}\}.$$

Lemma 2.2 (Yang [18])

Let $\{X_n,n\ge 1\}$ be a sequence of ψ-mixing random variables with $E{X}_k=0$, $|X_k|\le d<\mathrm{\infty }$ a.s., $k=1,2,\dots$ , $0<\lambda <1$, $m=[n^{\lambda }]$. Then $\mathrm{\forall }\epsilon >0$,

$$P(\left|\sum _{k=1}^n{X}_k\right|>\epsilon )\le 2e{C}_{1}exp\{-t\epsilon +C_2{t}^2{B}_n\},$$

where $B_n={\sum }_{k=1}^{n}E{X}_k^2$, $tmd\le 1/4$, $C_1=exp\{2e{n}^{1-\lambda }\psi (m)\}$, $C_2=4(1+4{\sum }_{k=1}^{2m}\psi (k))$.

Lemma 2.3 Let $\{X_n,n\ge 1\}$ be a sequence of ψ-mixing random variables, and let $A_j=\{|X_j|\ge x_j\}$, $x_j\in {\mathbb{R}}^{+}$, $j=1,2,\dots ,N$, then

$$P(A_1,A_2,\dots ,A_N)\le {(1+\psi (1))}^N\prod _{j=1}^{N}P(A_j).$$

Proof By the definition of ψ-mixing, we have

$$\begin{array}{rcl}P(A_1,A_2,\dots ,A_N)& \le & (1+\psi (1))P(A_1)P(A_2,\dots ,A_N)\\ \le & \cdots \\ \le & {(1+\psi (1))}^{N}P(A_1)P(A_2)\cdots P(A_N).\end{array}$$

The proof is complete. □

3 Proofs

In this section, we state the proofs of our main results.

Proof of Theorem 2.1 Let $S_{nj}={\sum }_{k=1}^j{a}_{nk}{X}_{nk}$, $1\le j\le n$. Since $a_{nk}=a_{nk}^{+}-a_{nk}^{-}$, without loss of generality, we may assume that $0<a_{nk}\le C{n}^{-\alpha }$. Let $0<\rho <\frac{(2\alpha p-1)(N-1)}{2pN}$, where N is a positive integer with $N>1$. Let

$$\begin{array}{c}{X}_{nk}^{\mathrm{\prime }}=X_{nk}{I}_{(a_{nk}|X_{nk}|\le n^{-\rho })},X_{nk}^{\mathrm{\prime }\mathrm{\prime }}=X_{nk}{I}_{(a_{nk}|X_{nk}|>\epsilon /N)},\hfill \\ X_{nk}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=X_k-X_{nk}^{\mathrm{\prime }}-X_{nk}^{\mathrm{\prime }\mathrm{\prime }}=X_{nk}{I}_{(n^{-\rho }<a_{nk}|X_{nk}|\le \epsilon /N)},\hfill \\ S_{nj}^{\mathrm{\prime }}=\sum _{k=1}^j{a}_{nk}{X}_{nk}^{\mathrm{\prime }},S_{nj}^{\mathrm{\prime }\mathrm{\prime }}=\sum _{k=1}^j{a}_{nk}{X}_{nk}^{\mathrm{\prime }\mathrm{\prime }},S_{nj}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=\sum _{k=1}^j{a}_{nk}{X}_{nk}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}.\hfill \end{array}$$

Firstly, we prove ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P({max}_{1\le j\le n}|S_{nj}^{\mathrm{\prime }}|>\epsilon )<\mathrm{\infty }$. By $\{X_n\}\prec X$, we know that $E{|X_{nk}|}^{2p}\le E{X}^{2p}<\mathrm{\infty }$. If $0<p\le 1/2$, we have

$$\begin{array}{rl}\underset{1\le j\le n}{max}\left|\sum _{k=1}^j{a}_{nk}E{X}_{nk}^{\mathrm{\prime }}\right|& \le \sum _{k=1}^n{a}_{nk}\left|E{X}_{nk}^{\mathrm{\prime }}\right|\le \sum _{k=1}^n{a}_{nk}E|X_{nk}|I_{(a_{nk}|X_{nk}|\le n^{-\rho })}\\ \le \sum _{k=1}^n\frac{a_{nk}^{2p}E{|X_{nk}|}^{2p}}{n^{-2\rho p}}{n}^{-\rho }\le C{n}^{\rho (2p-1)-(2\alpha p-1)}\to 0,n\to \mathrm{\infty }.\end{array}$$

(1)

If $p>1/2$, by $E{X}_{nk}=0$ and $\rho <\frac{2\alpha p-1}{2p}<\frac{2\alpha p-1}{2p-1}$, we also have

$$\begin{array}{rcl}\underset{1\le j\le n}{max}\left|\sum _{k=1}^j{a}_{nk}E{X}_{nk}^{\mathrm{\prime }}\right|& \le & \sum _{k=1}^n{a}_{nk}\left|E{X}_{nk}^{\mathrm{\prime }}\right|\le \sum _{k=1}^n{a}_{nk}E|X_{nk}|I_{(a_{nk}|X_{nk}|>n^{-\rho })}\\ \le & \sum _{k=1}^n\frac{a_{nk}^{2p}E{|X_{nk}|}^{2p}}{n^{-2\rho p}}{n}^{-\rho }\le C{n}^{\rho (2p-1)-(2\alpha p-1)}\to 0,n\to \mathrm{\infty }.\end{array}$$

Therefore, we know that (1) holds for $p>0$. Let $S_{nj}^{\ast }={\sum }_{k=1}^j{a}_{nk}(X_{nk}^{\mathrm{\prime }}-E{X}_{nk}^{\mathrm{\prime }})$. To prove ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P({max}_{1\le j\le n}|S_{nj}^{\mathrm{\prime }}|>\epsilon )<\mathrm{\infty }$, it suffices to show that ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P({max}_{1\le j\le n}|S_{nj}^{\ast }|>\epsilon )<\mathrm{\infty }$.

If $0<p<1$, by Markov’s inequality and Lemma 2.1, we have

$$\begin{array}{r}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\ast }\right|>\epsilon )\\ \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}E{\left(\underset{1\le j\le n}{max}\left|S_{nj}^{\ast }\right|\right)}^2\le C\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}\sum _{k=1}^n{a}_{nk}^{2}E{\left(X_{nk}^{\mathrm{\prime }}\right)}^2\\ \le \sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}\sum _{k=1}^n{a}_{nk}^{2p}E\left({\left|a_{nk}{X}_{nk}^{\mathrm{\prime }}\right|}^{2-2p}{\left|X_{nk}^{\mathrm{\prime }}\right|}^{2p}\right)\\ \le \sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}{n}^{-\rho (2-2p)}\sum _{k=1}^n{a}_{nk}^{2p}E{\left|X_{nk}^{\mathrm{\prime }}\right|}^{2p}\le C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1-2\rho (1-p)}<\mathrm{\infty }.\end{array}$$

(2)

If $p\ge 1$, take $q>max\{2p,2(2\alpha p-1)/\theta \}$. By $q>2$ and Lemma 2.1, we have

$$\begin{array}{r}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\ast }\right|>\epsilon )\\ \le \sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}[\sum _{k=1}^n{a}_{nk}^{q}E{\left|X_{nk}^{\mathrm{\prime }}\right|}^q+{\left(\sum _{k=1}^n{a}_{nk}^{2}E{\left(X_{nk}^{\mathrm{\prime }}\right)}^2\right)}^{q/2}].\end{array}$$

(3)

By a similar argument as in the proof of (2) (replacing exponent 2 into q), we can get

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}\sum _{k=1}^n{a}_{nk}^{q}E{\left|X_{nk}^{\mathrm{\prime }}\right|}^q<\mathrm{\infty }.$$

(4)

Note that $E{|X_{nk}^{\mathrm{\prime }}|}^2\le E{X}^2<\mathrm{\infty }$ and the definition of q, we have

$$\begin{array}{r}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}{\left(\sum _{k=1}^n{a}_{nk}^{2}E{\left(X_{nk}^{\mathrm{\prime }}\right)}^2\right)}^{q/2}\\ \le \sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}{\left(\sum _{k=1}^n{a}_{nk}^2\right)}^{q/2}{\left(E{X}^2\right)}^{q/2}\le C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1+(2\alpha p-1)-\theta q/2}{\left(E{X}^2\right)}^{q/2}<\mathrm{\infty }.\end{array}$$

(5)

From (3)-(5), we know that (2) still holds for $p\ge 1$. By (1) and (2), we have

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }}\right|>\epsilon )<\mathrm{\infty }.$$

Secondly, we prove ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P({max}_{1\le j\le n}|S_{nj}^{\mathrm{\prime }\mathrm{\prime }}|>\epsilon )<\mathrm{\infty }$. Let ${\phi }_n(j)=\mathrm{♯}\{1\le k\le n:a_{nk}>\epsilon /(jN)\}$ and ${\varphi }_j=[{(jCN/\epsilon )}^{1/\alpha }]$, then

$$\begin{array}{rcl}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )& \le & P\left(\bigcup _{k=1}^n\{a_{nk}|X_{nk}|>\epsilon /N\}\right)\\ \le & \sum _{k=1}^{n}P(a_{nk}|X_{nk}|>\epsilon /N)\le C\sum _{k=1}^{n}P(a_{nk}X>\epsilon /N)\\ =& C\sum _{j=1}^{\mathrm{\infty }}\sum _{k=1}^{n}P(a_{nk}X>\epsilon /N,j-1\le X<j)\\ \le & C\sum _{j=1}^{\mathrm{\infty }}{\phi }_n(j)P(j-1\le X<j).\end{array}$$

(6)

By $a_{nk}X>\epsilon /N$, we know $a_{nk}>\epsilon /(jN)$. Note $a_{nk}\le C{n}^{-\alpha }$, we have $n<{(jCN/\epsilon )}^{1/\alpha }$. Hence, we have $n\le {\varphi }_j$, then

$$\begin{array}{rcl}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )& \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}\sum _{j=1}^{\mathrm{\infty }}{\phi }_n(j)P(j-1\le X<j)\\ \le & C\sum _{j=1}^{\mathrm{\infty }}\sum _{n=1}^{{\varphi }_j}{n}^{2\alpha p-2}{\phi }_n(j)P(j-1\le X<j).\end{array}$$

(7)

Take $v\in (0,\frac{2\alpha p-1}{\alpha })$, then ${\sum }_{k=1}^n{a}_{nk}^v\ge {\phi }_n(j){\epsilon }^v/{(jN)}^v$. From ${\sum }_{k=1}^n{a}_{nk}^v\le C{n}^{1-\alpha v}$, we have

$${\phi }_n(j)\le N^v{j}^v/{\epsilon }^v\sum _{k=1}^n{a}_{nk}^v\le C{n}^{1-\alpha v}{j}^v.$$

(8)

Therefore, by (7) and (8), we have

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )\le C\sum _{j=1}^{\mathrm{\infty }}\sum _{n=1}^{{\varphi }_j}{n}^{2\alpha p-\alpha v-1}{j}^{v}P(j-1\le X<j).$$

(9)

By the definition of v, we have $2\alpha p-\alpha v-1>0$, then

$$\sum _{n=1}^{{\varphi }_j}{n}^{2\alpha p-\alpha v-1}\le {\varphi }_j^{\alpha (2p-v)}\le C{j}^{2p-v}.$$

(10)

From (9), (10) and $E{X}^{2p}<\mathrm{\infty }$, then

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )\le C\sum _{j=1}^{\mathrm{\infty }}{j}^{2p}P(j-1\le X<j)<\mathrm{\infty }.$$

Finally, we prove ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P({max}_{1\le j\le n}|S_{nj}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}|>\epsilon )<\mathrm{\infty }$. Obviously, we know that $P({max}_{1\le j\le n}|S_{nj}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}|>\epsilon )\le P({\sum }_{k=1}^n{a}_{nk}|X_{nk}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}|>\epsilon )$. Let $M=\mathrm{♯}\{1\le k\le n:n^{-\rho }<a_{nk}|X_{nk}|\le \epsilon /N\}$. We must let M be at least N such that ${\sum }_{k=1}^n{a}_{nk}|X_{nk}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}|>\epsilon$. Take $W_k=\{a_{nk}|X_{nk}|>n^{-\rho }\}$, we have

$$\begin{array}{rcl}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )& \le & P(M\text{ is at least }N,\text{ such that }{a}_{nk}|X_{nk}|>n^{-\rho })\\ \le & \sum _{1\le i_1<i_2<\cdots <i_N\le n}P(W_{i_1}{W}_{i_2}\cdots W_{i_N}).\end{array}$$

(11)

By Lemma 2.3, we have

$$P(W_{i_1}{W}_{i_2}\cdots W_{i_N})\le {(1+\psi (1))}^{N}P(W_{i_1})P(W_{i_2})\cdots P(W_{i_N}).$$

(12)

From (11) and (12), we have

$$\begin{array}{r}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )\\ \le {(1+\psi (1))}^N{n}^{2\alpha p-2}\sum _{1\le i_1<i_2<\cdots <i_N\le n}P(W_{i_1})P(W_{i_2})\cdots P(W_{i_N})\\ \le C{n}^{2\alpha p-2}\left(\begin{array}{c}n\\ N\end{array}\right)P^N(X>a_{nk}^{-1}{n}^{-\rho })\\ \le C{n}^{2\alpha p-2}{n}^N{P}^N(X>C^{-1}{n}^{-\rho +\alpha })\\ \le C{n}^{-1-(2\alpha p-1)(N-1)+2\rho pN}{\left(E{X}^{2p}\right)}^N.\end{array}$$

(13)

Noting that $0<\rho <\frac{(2\alpha p-1)(N-1)}{2pN}$. We have

$$-(2\alpha p-1)(N-1)+2\rho pN<0,$$

then

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )\le C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1-(2\alpha p-1)(N-1)+2\rho pN}<\mathrm{\infty }.$$

The proof is completed. □

Proof of Theorem 2.2 Following the notations of $X_{nk}^{\mathrm{\prime }}$, $X_{nk}^{\mathrm{\prime }\mathrm{\prime }}$ and $X_{nk}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}$, but let

$$\begin{array}{c}{T}_n^{\mathrm{\prime }}=\sum _{k=1}^n{a}_{nk}{X}_{nk}^{\mathrm{\prime }},T_n^{\mathrm{\prime }\mathrm{\prime }}=\sum _{k=1}^n{a}_{nk}{X}_{nk}^{\mathrm{\prime }\mathrm{\prime }},\hfill \\ T_n^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=\sum _{k=1}^n{a}_{nk}{X}_{nk}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }},T_n^{\ast }=\sum _{k=1}^n{a}_{nk}(X_{nk}^{\mathrm{\prime }}-E{X}_{nk}^{\mathrm{\prime }}).\hfill \end{array}$$

Obviously, by following the methods used in the proof of (1), we have

$$\left|\sum _{k=1}^n{a}_{nk}E{X}_{nk}^{\mathrm{\prime }}\right|\to 0,n\to \mathrm{\infty }.$$

(14)

By similar arguments as in the proofs of

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )<\mathrm{\infty }\text{and}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\underset{1\le j\le n}{max}\left|S_{nj}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )<\mathrm{\infty },$$

we can prove

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\left|T_n^{\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )<\mathrm{\infty }\text{and}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\left|T_n^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\right|>\epsilon )<\mathrm{\infty }.$$

Here, we omit the details. Therefore, we need only to show

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\left|T_n^{\ast }\right|>\epsilon )<\mathrm{\infty }.$$

Take $\lambda <\rho <\frac{(2\alpha p-1)(N-1)}{2pN}$, by $0<\lambda <\frac{1}{2p}$ and $p\ge 1$, we know $0<\frac{\lambda }{1-\lambda }<1$. Hence, from condition (i), we have

$$C_2=4(1+4\sum _{k=1}^{2m}\psi (k))\le 4(1+4\sum _{k=1}^{\mathrm{\infty }}\psi (k))<\mathrm{\infty },\psi (m)=o\left(m^{\frac{\lambda -1}{\lambda }}\right),$$

where $m=[n^{\lambda }]$. Therefore, $C_1=exp\{2e{n}^{1-\lambda }\psi (m)\}\ll exp\{2e\}<\mathrm{\infty }$.

Take $t=\frac{2\alpha plogn}{\epsilon }$. Clearly, $t\le n^{\rho -\lambda }/8$ when n is sufficiently large. Note that $|a_{nk}(X_{nk}^{\mathrm{\prime }}-E{X}_{nk}^{\mathrm{\prime }})|\le 2n^{-\rho }=d$, then $tmd\le 1/4$ when n is sufficiently large. By Lemma 2.2, we have

$$\begin{array}{rcl}P(\left|T_n^{\ast }\right|>\epsilon )& \le & Cexp\{-t\epsilon +C_0{C}_2{t}^2\sum _{k=1}^n{a}_{nk}^2\}\\ =& Cexp\{-2\alpha plogn+C_0{C}_2\frac{{(2\alpha p)}^2{log}^{2}n}{{\epsilon }^2}\sum _{k=1}^n{a}_{nk}^2\},\end{array}$$

(15)

where $C_0=E{(X_{nk}^{\mathrm{\prime }})}^2\le E{X}^{2p}<\mathrm{\infty }$. Note that $logn{\sum }_{k=1}^n{a}_{nk}^2=o(1)$ holds if $\alpha >\frac{1}{2}$. Therefore, by condition (ii), we have

$$C_0{C}_2\frac{{(2\alpha p)}^{2}logn}{{\epsilon }^2}\sum _{k=1}^n{a}_{nk}^2\to 0,n\to \mathrm{\infty }.$$

(16)

Hence, when n is sufficiently large, by (14) and (15), we have

$$P(\left|T_n^{\ast }\right|>\epsilon )\le Cexp(-2\alpha plogn+2^{-1}logn)=C{n}^{-2\alpha p+1/2}.$$

Then

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{2\alpha p-2}P(\left|T_n^{\ast }\right|>\epsilon )\le C\sum _{n=1}^{\mathrm{\infty }}{n}^{-\frac{3}{2}}<\mathrm{\infty }.$$

The proof is completed. □