Complete convergence for weighted sums of arrays of rowwise $\tilde{\rho }$-mixing random variables

Abstract

Let $\{X_{ni},i\ge 1,n\ge 1\}$ be an array of rowwise $\tilde{\rho }$-mixing random variables. Some sufficient conditions for complete convergence for weighted sums of arrays of rowwise $\tilde{\rho }$-mixing random variables are presented without assumptions of identical distribution. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of $\tilde{\rho }$-mixing random variables are obtained.

MSC:60F15.

1 Introduction

The concept of complete convergence was introduced by Hsu and Robbins [1] as follows. A sequence of random variables $\{U_n,n\ge 1\}$ is said to converge completely to a constant C if ${\sum }_{n=1}^{\mathrm{\infty }}P(|U_n-C|>\epsilon )<\mathrm{\infty }$ for all $\epsilon >0$. In view of the Borel-Cantelli lemma, this implies that $U_n\to C$ almost surely (a.s.). The converse is true if the $\{U_n,n\ge 1\}$ are independent. Hsu and Robbins [1] proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Erdös [2] proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. See, for example, Spitzer [3], Baum and Katz [4], Gut [5], Zarei [6], and so forth. The main purpose of the paper is to provide complete convergence for weighted sums of arrays of rowwise $\tilde{\rho }$-mixing random variables.

Firstly, let us recall the definitions of sequences of $\tilde{\rho }$-mixing random variables and arrays of rowwise $\tilde{\rho }$-mixing random variables.

Let $\{X_n,n\ge 1\}$ be a sequence of random variables defined on a fixed probability space $(\mathrm{\Omega },\mathcal{F},P)$. Write $\mathcal{FS}=\sigma (X_i,i\in S\subset \mathbb{N})$. Given two σ-algebras ℬ, ℛ in ℱ, let

$$\rho (\mathcal{B},\mathcal{R})=\underset{X\in L_2(\mathcal{B}),Y\in L_2(\mathcal{R})}{sup}\frac{|EXY-EXEY|}{{(VarXVarY)}^{1/2}}.$$

Define the $\tilde{\rho }$-mixing coefficients by

$$\tilde{\rho }(k)=sup\{\rho (\mathcal{FS},\mathcal{FT}):\text{ finite subsets }S,T\subset \mathbb{N}\text{ such that }dist(S,T)\ge k\},k\ge 0.$$

Obviously, $0\le \tilde{\rho }(k+1)\le \tilde{\rho }(k)\le 1$, and $\tilde{\rho }(0)=1$.

Definition 1.1 A sequence $\{X_n,n\ge 1\}$ of random variables is said to be a $\tilde{\rho }$-mixing sequence if there exists $k\in \mathbb{N}$ such that $\tilde{\rho }(k)<1$.

An array $\{X_{ni},i\ge 1,n\ge 1\}$ of random variables is called rowwise $\tilde{\rho }$-mixing random variables if for every $n\ge 1$, $\{X_{ni},i\ge 1\}$ is a sequence of $\tilde{\rho }$-mixing random variables.

$\tilde{\rho }$-mixing random variables were introduced by Bradley [7] and many applications have been found. $\tilde{\rho }$-mixing is similar to ρ-mixing, but both are quite different. Many authors have studied this concept and provided interesting results and applications. See, for example, Bryc and Smolenski [8], Peligrad [9, 10], Peligrad and Gut [11], Utev and Peligrad [12], Gan [13], Cai [14], Zhu [15], Wu and Jiang [16, 17], An and Yuan [18], Kuczmaszewska [19], Sung [20], Wang et al. [21–23], and so on.

Recently, An and Yuan [18] obtained a complete convergence result for weighted sums of identically distributed $\tilde{\rho }$-mixing random variables as follows.

Theorem 1.1 Let $p>1/\alpha$ and $1/2<\alpha \le 2$. Let $\{X_n,n\ge 1\}$ be a sequence of identically distributed $\tilde{\rho }$-mixing random variables with $E{X}_1=0$. Assume that $\{a_{ni},1\le i\le n,n\ge 1\}$ is an array of real numbers satisfying

$$\sum _{i=1}^n{|a_{ni}|}^p=O\left(n^{\delta }\right)\mathit{\text{for some}}0<\delta <1,$$

(1.1)

$$\mathrm{♯}{A}_{nk}=\mathrm{♯}\{1\le i\le n:{|a_{ni}|}^p>{(k+1)}^{-1}\}\ge n{e}^{-1/k},\mathrm{\forall }k\ge 1,n\ge 1.$$

(1.2)

Then the following statements are equivalent:

• (i) $E{|X_1|}^p<\mathrm{\infty }$;

• (ii) ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{p\alpha -2}P({max}_{1\le j\le n}|{\sum }_{i=1}^j{a}_{ni}{X}_i|>\epsilon n^{\alpha })<\mathrm{\infty }$ for all $\epsilon >0$.

Sung [20] pointed out that the array $\{a_{ni},1\le i\le n,n\ge 1\}$ satisfying both (1.1) and (1.2) does not exist and obtained a new complete convergence result for weighted sums of identically distributed $\tilde{\rho }$-mixing random variables as follows.

Theorem 1.2 Let $p>1/\alpha$ and $1/2<\alpha \le 2$. Let $\{X_n,n\ge 1\}$ be a sequence of identically distributed $\tilde{\rho }$-mixing random variables with $E{X}_1=0$. Assume that $\{a_{ni},1\le i\le n,n\ge 1\}$ is an array of real numbers satisfying

$$\sum _{i=1}^n{|a_{ni}|}^q=O(n)\mathit{\text{for some}}q>p.$$

(1.3)

If $E{|X_1|}^p<\mathrm{\infty }$, then

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

(1.4)

Conversely, if (1.4) holds for any array $\{a_{ni},1\le i\le n,n\ge 1\}$ satisfying (1.3), then $E{|X_1|}^p<\mathrm{\infty }$.

For more details about the complete convergence result for weighted sums of dependent sequences, one can refer to Wu [24, 25], Wang et al. [26, 27], and so forth. The main purpose of this paper is to further study the complete convergence for weighted sums of arrays of rowwise $\tilde{\rho }$-mixing random variables under mild conditions. The main idea is inspired by Baek et al. [28] and Wu [25]. As applications, the results of Baum and Katz [4] from the i.i.d. case to the arrays of rowwise $\tilde{\rho }$-mixing setting are obtained. The Marcinkiewicz-Zygmund type strong law of large numbers for sequences of $\tilde{\rho }$-mixing random variables is provided. We give some sufficient conditions for complete convergence for weighted sums of arrays of rowwise $\tilde{\rho }$-mixing random variables without assumption of identical distribution. The techniques used in the paper are the Rosenthal type inequality and the truncation method.

Throughout this paper, the symbol C denotes a positive constant which is not necessarily the same one in each appearance and $\lceil x\rceil$ denotes the integer part of x. For a finite set A, the symbol ♯A denotes the number of elements in the set A. Let $I(A)$ be the indicator function of the set A. Denote $logx=lnmax(x,e)$, $X^{+}=max(X,0)$ and $X^{-}=max(-X,0)$.

The paper is organized as follows. Two important lemmas are provided in Section 2. The main results and their proofs are presented in Section 3. We get complete convergence for arrays of rowwise $\tilde{\rho }$-mixing random variables which are stochastically dominated by a random variable X.

2 Preliminaries

Firstly, we give the definition of stochastic domination.

Definition 2.1 A sequence $\{X_n,n\ge 1\}$ of random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C such that

$$P(|X_n|>x)\le CP(|X|>x)$$

(2.1)

for all $x\ge 0$ and $n\ge 1$.

An array $\{X_{ni},i\ge 1,n\ge 1\}$ of rowwise random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C such that

$$P(|X_{ni}|>x)\le CP(|X|>x)$$

(2.2)

for all $x\ge 0$, $i\ge 1$ and $n\ge 1$.

The proofs of the main results of the paper are based on the following two lemmas. One is the classic Rosenthal type inequality for $\tilde{\rho }$-mixing random variables obtained by Utev and Peligrad [12], the other is the fundamental inequalities for stochastic domination.

Lemma 2.1 (cf. Utev and Peligrad [[12], Theorem 2.1])

Let $\{X_n,n\ge 1\}$ be a sequence of $\tilde{\rho }$-mixing random variables, $E{X}_i=0$, $E{|X_i|}^p<\mathrm{\infty }$ for some $p\ge 2$ and for every $i\ge 1$. Then there exists a positive constant C depending only on p such that

$$E\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_i{|}^p)\le C\{\sum _{i=1}^{n}E{|X_i|}^p+{\left(\sum _{i=1}^{n}E{X}_i^2\right)}^{p/2}\}.$$

Lemma 2.2 Let $\{X_{ni},i\ge 1,n\ge 1\}$ be an array of rowwise random variables which is stochastically dominated by a random variable X. For any $\alpha >0$ and $b>0$, the following two statements hold:

$$E{|X_{ni}|}^{\alpha }I(|X_{ni}|\le b)\le C_1[E{|X|}^{\alpha }I(|X|\le b)+b^{\alpha }P(|X|>b)],$$

(2.3)

$$E{|X_{ni}|}^{\alpha }I(|X_{ni}|>b)\le C_{2}E{|X|}^{\alpha }I(|X|>b),$$

(2.4)

where $C_1$ and $C_2$ are positive constants.

Proof The proof of this lemma can be found in Wu [29] or Wang et al. [30]. □

3 Main results and their applications

In this section, we provide complete convergence for weighted sums of arrays of rowwise $\tilde{\rho }$-mixing random variables. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of $\tilde{\rho }$-mixing random variables are obtained. Let $\{X_{ni},i\ge 1,n\ge 1\}$ be an array of rowwise $\tilde{\rho }$-mixing random variables. We assume that the mixing coefficients $\tilde{\rho }(\cdot )$ in each row are the same.

Theorem 3.1 Let $\{X_{ni},i\ge 1,n\ge 1\}$ be an array of rowwise $\tilde{\rho }$-mixing random variables which is stochastically dominated by a random variable X and $E{X}_{ni}=0$ for all $i\ge 1$, $n\ge 1$, $\beta \ge -1$. Let $\{a_{ni},i\ge 1,n\ge 1\}$ be an array of constants such that

$$\underset{i\ge 1}{sup}|a_{ni}|=O\left(n^{-r}\right)\mathit{\text{for some}}r>0$$

(3.1)

and

$$\sum _{i=1}^{\mathrm{\infty }}|a_{ni}|=O\left(n^{\alpha }\right)\mathit{\text{for some}}\alpha \in [0,r).$$

(3.2)

Assume further that $1+\alpha +\beta >0$ and there exists some $\delta >0$ such that $\alpha /r+1<\delta \le 2$ and $s=max(1+\frac{1+\alpha +\beta }{r},\delta )$. If $E{|X|}^s<\mathrm{\infty }$, then for all $\epsilon >0$,

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{a}_{ni}{X}_{ni}|>\epsilon )<\mathrm{\infty }.$$

(3.3)

If $1+\alpha +\beta <0$ and $E|X|<\mathrm{\infty }$, then (3.3) still holds for all $\epsilon >0$.

Proof Without loss of generality, we assume that $a_{ni}>0$ for all $i\ge 1$ and $n\ge 1$ (otherwise, we use $a_{ni}^{+}$ and $a_{ni}^{-}$ instead of $a_{ni}$ respectively, and note that $a_{ni}=a_{ni}^{+}-a_{ni}^{-}$). From the conditions (3.1) and (3.2), we assume that

$$\underset{i\ge 1}{sup}{a}_{ni}=n^{-r},\sum _{i=1}^{\mathrm{\infty }}{a}_{ni}=n^{\alpha },n\ge 1.$$

(3.4)

If $1+\alpha +\beta <0$ and $E|X|<\mathrm{\infty }$, then the result can be easily proved by the following:

$$\begin{array}{rcl}\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{a}_{ni}{X}_{ni}|>\epsilon )& \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }E\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{a}_{ni}{X}_{ni}\left|\right)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}E|a_{ni}{X}_{ni}|\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha +\beta }E|X|<\mathrm{\infty }.\end{array}$$

In the following, we consider the case of $1+\alpha +\beta >0$. Denote

$$X_{ni}^{\prime }=a_{ni}{X}_{ni}I(|a_{ni}{X}_{ni}|\le 1),i\ge 1,n\ge 1.$$

It is easy to check that for any $\epsilon >0$,

$$\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{a}_{ni}{X}_{ni}|>\epsilon )\subset (\underset{1\le i\le n}{max}|a_{ni}{X}_{ni}|>1)\cup \left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_{ni}^{\prime }|>\epsilon ),$$

which implies that

$$\begin{array}{r}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{a}_{ni}{X}_{ni}|>\epsilon )\\ \le P(\underset{1\le i\le n}{max}|a_{ni}{X}_{ni}|>1)+P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_{ni}^{\prime }|>\epsilon )\\ \le \sum _{i=1}^{n}P(|a_{ni}{X}_{ni}|>1)+P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j(X_{ni}^{\prime }-E{X}_{ni}^{\prime })|>\epsilon -\underset{1\le j\le n}{max}|\sum _{i=1}^{j}E{X}_{ni}^{\prime }\left|\right).\end{array}$$

(3.5)

Firstly, we show that

$$\underset{1\le j\le n}{max}\left|\sum _{i=1}^{j}E{X}_{ni}^{\prime }\right|\to 0\text{as }n\to \mathrm{\infty }.$$

(3.6)

Actually, by the conditions $E{X}_{ni}=0$, Lemma 2.2, (3.4) and $E{|X|}^{1+\alpha /r}<\mathrm{\infty }$ (since $E{|X|}^s<\mathrm{\infty }$), we have that

$$\begin{array}{rcl}\underset{1\le j\le n}{max}\left|\sum _{i=1}^{j}E{X}_{ni}^{\prime }\right|& =& \underset{1\le j\le n}{max}\left|\sum _{i=1}^{j}E{a}_{ni}{X}_{ni}I(|a_{ni}{X}_{ni}|\le 1)\right|\\ =& \underset{1\le j\le n}{max}\left|\sum _{i=1}^{j}E{a}_{ni}{X}_{ni}I(|a_{ni}{X}_{ni}|>1)\right|\\ \le & \sum _{i=1}^{n}E|a_{ni}{X}_{ni}{|}^{1+\alpha /r}I(x|a_{ni}{X}_{ni}|>1)\\ \le & C\sum _{i=1}^n{a}_{ni}^{1+\alpha /r}E{|X|}^{1+\alpha /r}I(|X|>\frac{1}{a_{ni}})\\ \le & C{\left(\underset{i\ge 1}{sup}{a}_{ni}\right)}^{\alpha /r}\sum _{i=1}^n{a}_{ni}E{|X|}^{1+\alpha /r}I(|X|>n^r)\\ \le & C{\left(n^{-r}\right)}^{\alpha /r}{n}^{\alpha }E{|X|}^{1+\alpha /r}I(|X|>n^r)\\ =& CE{|X|}^{1+\alpha /r}I(|X|>n^r)\to 0\text{as }n\to \mathrm{\infty },\end{array}$$

which implies (3.6). It follows from (3.5) and (3.6) that for n large enough,

$$P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{a}_{ni}{X}_{ni}|>\epsilon )\le \sum _{i=1}^{n}P(|a_{ni}{X}_{ni}|>1)+P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j(X_{ni}^{\prime }-E{X}_{ni}^{\prime })|>\frac{\epsilon }{2}).$$

Hence, to prove (3.3), we only need to show that

$$I\doteq \sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}P(|a_{ni}{X}_{ni}|>1)<\mathrm{\infty }$$

(3.7)

and

$$J\doteq \sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j(X_{ni}^{\prime }-E{X}_{ni}^{\prime })|>\frac{\epsilon }{2})<\mathrm{\infty }.$$

(3.8)

By (3.4) and $E{|X|}^s<\mathrm{\infty }$, we can get that

$$\begin{array}{rcl}\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}P(|a_{ni}{X}_{ni}|>1)& \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}P(|a_{ni}X|>1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^n{a}_{ni}E|X|I(|X|>\frac{1}{a_{ni}})\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha +\beta }E|X|I(|X|>n^r)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha +\beta }\sum _{k=n}^{\mathrm{\infty }}E|X|I(k^r\le |X|<{(k+1)}^r)\\ =& C\sum _{k=1}^{\mathrm{\infty }}\sum _{n=1}^k{n}^{\alpha +\beta }E|X|I(k^r\le |X|<{(k+1)}^r)\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^{1+\alpha +\beta }E|X|I(k^r\le |X|<{(k+1)}^r)\\ \le & C\sum _{k=1}^{\mathrm{\infty }}E{|X|}^{1+(1+\alpha +\beta )/r}I(k^r\le |X|<{(k+1)}^r)\\ \le & CE{|X|}^{1+(1+\alpha +\beta )/r}<\mathrm{\infty },\end{array}$$

which implies (3.7).

By Markov’s inequality, Lemma 2.1, $C_r$’s inequality and Jensen’s inequality, we have for $M\ge 2$ that

$$\begin{array}{r}\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j(X_{ni}^{\prime }-E{X}_{ni}^{\prime })|>\frac{\epsilon }{2})\\ \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }E\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j(X_{ni}^{\prime }-E{X}_{ni}^{\prime }){|}^M)\\ \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }[{\left(\sum _{i=1}^{n}E{\left|X_{ni}^{\prime }\right|}^2\right)}^{M/2}+\sum _{i=1}^{n}E{\left|X_{ni}^{\prime }\right|}^M]\\ \doteq J_1+J_2.\end{array}$$

(3.9)

Take

$$M>max(2,\frac{2(1+\beta )}{r[\delta -(1+\alpha /r)]},1+\frac{1+\alpha +\beta }{r}),$$

which implies that $\beta -r[\delta -(1+\alpha /r)]M/2<-1$ and $\alpha +\beta -r(M-1)<-1$. Since $E{|X|}^{\delta }<\mathrm{\infty }$, we have by Lemma 2.2, Markov’s inequality and (3.4) that

$$\begin{array}{rcl}{J}_1& \doteq & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }{\left(\sum _{i=1}^{n}E{\left|X_{ni}^{\prime }\right|}^2\right)}^{M/2}\\ =& C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }{\left[\sum _{i=1}^{n}E{|a_{ni}{X}_{ni}|}^{2}I(|a_{ni}{X}_{ni}|\le 1)\right]}^{M/2}\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }{[\sum _{i=1}^{n}P(|a_{ni}X|>1)+\sum _{i=1}^{n}E{|a_{ni}X|}^{2}I(|a_{ni}X|\le 1)]}^{M/2}\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }{\left(\sum _{i=1}^n{a}_{ni}^{\delta }E{|X|}^{\delta }\right)}^{M/2}(\text{since }\delta \le 2)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }{\left[{\left(\underset{i\ge 1}{sup}{a}_{ni}\right)}^{\delta -1}\sum _{i=1}^n{a}_{ni}\right]}^{M/2}\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }{[n^{-r(\delta -1)}\cdot n^{\alpha }]}^{M/2}\\ =& C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta -r[\delta -(1+\alpha /r)]M/2}<\mathrm{\infty }.\end{array}$$

(3.10)

By Lemma 2.2 again, we can see that

$$\begin{array}{rcl}{J}_2& \doteq & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}E{\left|X_{ni}^{\prime }\right|}^M\\ =& C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}E{|a_{ni}{X}_{ni}|}^{M}I(|a_{ni}{X}_{ni}|\le 1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}P(|a_{ni}X|>1)+C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}E{|a_{ni}X|}^{M}I(|a_{ni}X|\le 1)\\ \doteq & J_3+J_4.\end{array}$$

(3.11)

$J_3<\mathrm{\infty }$ has been proved by (3.7). In the following, we show that $J_4<\mathrm{\infty }$. Denote

$$I_{nj}=\{i:{(nj)}^r\le 1/a_{ni}<{[n(j+1)]}^r\},n\ge 1,j\ge 1.$$

(3.12)

It is easily seen that $I_{nk}\cap I_{nj}=\mathrm{\varnothing }$ for $k\ne j$ and ${\bigcup }_{j=1}^{\mathrm{\infty }}{I}_{nj}=\mathbb{N}$ for all $n\ge 1$. Hence,

$$\begin{array}{rcl}{J}_4& =& C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{i=1}^{n}E{|a_{ni}X|}^{M}I(|a_{ni}X|\le 1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{j=1}^{\mathrm{\infty }}\sum _{i\in I_{nj}}E{|a_{ni}X|}^{M}I(|a_{ni}X|\le 1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{j=1}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}E{|X|}^{M}I(|X|\le {[n(j+1)]}^r)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{j=1}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}\sum _{k=0}^{n(j+1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ =& C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{j=1}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}\sum _{k=0}^{2n}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ +C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{j=1}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}\sum _{k=2n+1}^{n(j+1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \doteq & J_5+J_6.\end{array}$$

(3.13)

It is easily seen that for all $m\ge 1$, we have that

$$\begin{array}{rcl}{n}^{\alpha }& =& \sum _{i=1}^{\mathrm{\infty }}{a}_{ni}=\sum _{j=1}^{\mathrm{\infty }}\sum _{i\in I_{nj}}{a}_{ni}\ge \sum _{j=1}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){[n(j+1)]}^{-r}\\ \ge & \sum _{j=m}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){[n(j+1)]}^{-r}\ge \sum _{j=m}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){[n(j+1)]}^{-r}{\left[\frac{n(m+1)}{n(j+1)}\right]}^{r(M-1)}\\ =& \sum _{j=m}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){[n(j+1)]}^{-rM}{[n(m+1)]}^{r(M-1)},\end{array}$$

which implies that for all $m\ge 1$,

$$\sum _{j=m}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}\le C{n}^{\alpha }\cdot n^{-r(M-1)}\cdot m^{-r(M-1)}=C{n}^{\alpha -r(M-1)}\cdot m^{-r(M-1)}.$$

(3.14)

Therefore,

$$\begin{array}{rcl}{J}_5& \doteq & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{j=1}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}\sum _{k=0}^{2n}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\cdot n^{\alpha -r(M-1)}\sum _{k=0}^{2n}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{k=0}^2\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha +\beta -r(M-1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ +C\sum _{k=2}^{\mathrm{\infty }}\sum _{n=\lceil k/2\rceil }^{\mathrm{\infty }}{n}^{\alpha +\beta -r(M-1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C+C\sum _{k=2}^{\mathrm{\infty }}{k}^{1+\alpha +\beta -r(M-1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C+C\sum _{k=2}^{\mathrm{\infty }}E{|X|}^{M+\frac{1+\alpha +\beta }{r}-(M-1)}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C+CE{|X|}^{1+\frac{1+\alpha +\beta }{r}}<\mathrm{\infty }(\text{since }E{|X|}^s<\mathrm{\infty })\end{array}$$

(3.15)

and

$$\begin{array}{rcl}{J}_6& \doteq & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{j=1}^{\mathrm{\infty }}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}\sum _{k=2n+1}^{n(j+1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{k=2n+1}^{\mathrm{\infty }}\sum _{j\ge \frac{k}{n}-1}(\mathrm{♯}{I}_{nj}){(nj)}^{-rM}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{\beta }\sum _{k=2n+1}^{\mathrm{\infty }}{n}^{\alpha -r(M-1)}{\left(\frac{k}{n}\right)}^{-r(M-1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{k=2}^{\mathrm{\infty }}\sum _{n=1}^{\lceil k/2\rceil }{n}^{\alpha +\beta }\cdot k^{-r(M-1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{k=2}^{\mathrm{\infty }}{k}^{1+\alpha +\beta -r(M-1)}E{|X|}^{M}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{k=2}^{\mathrm{\infty }}E{|X|}^{M+\frac{1+\alpha +\beta }{r}-(M-1)}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & CE{|X|}^{1+\frac{1+\alpha +\beta }{r}}<\mathrm{\infty }(\text{since}E{|X|}^s<\mathrm{\infty }).\end{array}$$

(3.16)

Thus, the inequality (3.8) follows from (3.9)-(3.11), (3.13), (3.15) and (3.16). This completes the proof of the theorem. □

Theorem 3.2 Let $\{X_{ni},i\ge 1,n\ge 1\}$ be an array of rowwise $\tilde{\rho }$-mixing random variables which is stochastically dominated by a random variable X and $E{X}_{ni}=0$ for all $i\ge 1$, $n\ge 1$. Let $\{a_{ni},i\ge 1,n\ge 1\}$ be an array of constants such that (3.1) holds and

$$\sum _{i=1}^{\mathrm{\infty }}|a_{ni}|=O(1).$$

(3.17)

If $E|X|log|X|<\mathrm{\infty }$, then for all $\epsilon >0$,

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{a}_{ni}{X}_{ni}|>\epsilon )<\mathrm{\infty }.$$

(3.18)

Proof We use the same notations as those in Theorem 3.1. According to the proof of Theorem 3.1, we only need to show that (3.7) and (3.8) hold, where $\beta =-1$ and $\alpha =0$.

The fact $E|X|log|X|<\mathrm{\infty }$ yields that

$$\begin{array}{rcl}I& \doteq & \sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\sum _{i=1}^{n}P(|a_{ni}{X}_{ni}|>1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\sum _{i=1}^{n}P(|a_{ni}X|>1)\\ \le & C\sum _{k=1}^{\mathrm{\infty }}\sum _{n=1}^k{n}^{-1}E|X|I(k^r\le |X|<{(k+1)}^r)\\ \le & C\sum _{k=1}^{\mathrm{\infty }}logkE|X|I(k^r\le |X|<{(k+1)}^r)\\ \le & C\sum _{k=1}^{\mathrm{\infty }}E|X|log|X|I(k^r\le |X|<{(k+1)}^r)\\ \le & CE|X|log|X|<\mathrm{\infty },\end{array}$$

which implies (3.7) for $\beta =-1$.

By Markov’s inequality, Lemmas 2.1 and 2.2, we can get that

$$\begin{array}{rcl}J& \doteq & \sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j(X_{ni}^{\prime }-E{X}_{ni}^{\prime })|>\frac{\epsilon }{2})\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\sum _{i=1}^{n}E{\left|X_{ni}^{\prime }\right|}^2\\ =& C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\sum _{i=1}^{n}E{|a_{ni}{X}_{ni}|}^{2}I(|a_{ni}{X}_{ni}|\le 1)\\ \le & C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\sum _{i=1}^{n}P(|a_{ni}X|>1)\\ +C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\sum _{i=1}^{n}E{|a_{ni}X|}^{2}I(|a_{ni}X|\le 1)\\ \le & C+J_5^{\ast }+J_6^{\ast }.\end{array}$$

(3.19)

Here, $J_5^{\ast }$ and $J_6^{\ast }$ are $J_5$ and $J_6$ when $M=2$, respectively. Similar to the proof of $J_5$, we can get that

$$J_5^{\ast }\le C+CE|X|<\mathrm{\infty }.$$

(3.20)

Similar to the proof of $J_6$, we have

$$\begin{array}{rcl}{J}_6^{\ast }& \le & C\sum _{k=2}^{\mathrm{\infty }}\sum _{n=1}^{[k/2]}{n}^{-1}\cdot k^{-r}E{|X|}^{2}I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & C\sum _{k=2}^{\mathrm{\infty }}logk\cdot k^{-r}\cdot k^{r}E|X|I(k\le {|X|}^{\frac{1}{r}}<k+1)\\ \le & CE|X|log|X|<\mathrm{\infty }.\end{array}$$

(3.21)

This completes the proof of the theorem from the statements above. □

By Theorems 3.1 and 3.2, we can extend the results of Baum and Katz [4] for independent and identically distributed random variables to the case of arrays of rowwise $\tilde{\rho }$-mixing random variables as follows.

Corollary 3.1 Let $\{X_{ni},i\ge 1,n\ge 1\}$ be an array of rowwise $\tilde{\rho }$-mixing random variables which is stochastically dominated by a random variable X and $E{X}_{ni}=0$ for all $i\ge 1$, $n\ge 1$.

• (i) Let $p>1$ and $1\le t<2$. If $E{|X|}^{pt}<\mathrm{\infty }$, then for all $\epsilon >0$,

  $$\sum _{n=1}^{\mathrm{\infty }}{n}^{p-2}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_{ni}|>\epsilon n^{\frac{1}{t}})<\mathrm{\infty }.$$

  (3.22)

• (ii) If $E|X|log|X|<\mathrm{\infty }$, then for all $\epsilon >0$,

  $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_{ni}|>\epsilon n)<\mathrm{\infty }.$$

  (3.23)

Proof (i) Let $a_{ni}=0$ if $i>n$ and $a_{ni}=n^{-1/t}$ if $i\le n$. Hence, conditions (3.1) and (3.2) hold for $r=1/t$ and $\alpha =1-1/t<r$. $\beta \doteq p-2>-1$. It is easy to check that

$$1+\alpha +\beta =p-\frac{1}{t}>0,1+\frac{1+\alpha +\beta }{r}=pt\doteq s,\frac{\alpha }{r}+1=t<pt\doteq s.$$

Therefore, the desired result (3.22) follows from Theorem 3.1 immediately.

• (ii) Let $a_{ni}=0$ if $i>n$ and $a_{ni}=n^{-1}$ if $i\le n$. Hence, conditions (3.1) and (3.17) hold for $r=-1$. Therefore, the desired result (3.23) follows from Theorem 3.2 immediately. This completes the proof of the corollary. □

Similar to the proofs of Theorems 3.1-3.2 and Corollary 3.1, we can get the following Baum and Katz type result for sequences of $\tilde{\rho }$-mixing random variables.

Theorem 3.3 Let $\{X_n,n\ge 1\}$ be a sequence of $\tilde{\rho }$-mixing random variables which is stochastically dominated by a random variable X and $E{X}_n=0$ for $n\ge 1$.

• (i) Let $p>1$ and $1\le t<2$. If $E{|X|}^{pt}<\mathrm{\infty }$, then for all $\epsilon >0$,

  $$\sum _{n=1}^{\mathrm{\infty }}{n}^{p-2}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_i|>\epsilon n^{\frac{1}{t}})<\mathrm{\infty }.$$

  (3.24)

• (ii) If $E|X|log|X|<\mathrm{\infty }$, then for all $\epsilon >0$,

  $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_i|>\epsilon n)<\mathrm{\infty }.$$

  (3.25)

By Theorem 3.3, we can get the Marcinkiewicz-Zygmund type strong law of large numbers for $\tilde{\rho }$-mixing random variables as follows.

Corollary 3.2 Let $\{X_n,n\ge 1\}$ be a sequence of $\tilde{\rho }$-mixing random variables which is stochastically dominated by a random variable X and $E{X}_n=0$ for $n\ge 1$.

• (i) Let $p>1$ and $1\le t<2$. If $E{|X|}^{pt}<\mathrm{\infty }$, then

  $$n^{-\frac{1}{t}}\sum _{i=1}^n{X}_i\to 0\mathit{\text{a.s.,}}n\to \mathrm{\infty }.$$

  (3.26)

• (ii) If $E|X|log|X|<\mathrm{\infty }$, then

  $$\frac{1}{n}\sum _{i=1}^n{X}_i\to 0\mathit{\text{a.s.,}}n\to \mathrm{\infty }.$$

  (3.27)

Proof (i) By (3.24), we can get that for all $\epsilon >0$,

$$\begin{array}{rcl}\mathrm{\infty }& >& \sum _{n=1}^{\mathrm{\infty }}{n}^{p-2}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_i|>\epsilon n^{\frac{1}{t}})\\ =& \sum _{k=0}^{\mathrm{\infty }}\sum _{n=2^k}^{2^{k+1}-1}{n}^{p-2}P\left(\underset{1\le j\le n}{max}\right|\sum _{i=1}^j{X}_i|>\epsilon n^{\frac{1}{t}})\\ \ge & \{\begin{array}{ll}{\sum }_{k=0}^{\mathrm{\infty }}{(2^k)}^{p-2}{2}^{k}P({max}_{1\le j\le 2^k}|{\sum }_{i=1}^j{X}_i|>\epsilon 2^{\frac{k+1}{t}})& \text{if }p\ge 2,\\ {\sum }_{k=0}^{\mathrm{\infty }}{(2^{k+1})}^{p-2}{2}^{k}P({max}_{1\le j\le 2^k}|{\sum }_{i=1}^j{X}_i|>\epsilon 2^{\frac{k+1}{t}})& \text{if }1<p<2,\end{array}\\ \ge & \{\begin{array}{ll}{\sum }_{k=0}^{\mathrm{\infty }}P({max}_{1\le j\le 2^k}|{\sum }_{i=1}^j{X}_i|>\epsilon 2^{\frac{k+1}{t}})& \text{if }p\ge 2,\\ \frac{1}{2}{\sum }_{k=0}^{\mathrm{\infty }}P({max}_{1\le j\le 2^k}|{\sum }_{i=1}^j{X}_i|>\epsilon 2^{\frac{k+1}{t}})& \text{if }1<p<2.\end{array}\end{array}$$

By Borel-Cantelli lemma, we obtain that

$$\frac{{max}_{1\le j\le 2^k}|{\sum }_{i=1}^j{X}_i|}{2^{\frac{k+1}{t}}}\to 0\text{a.s., }k\to \mathrm{\infty }.$$

(3.28)

For all positive integers n, there exists a positive integer $k_0$ such that $2^{k_0-1}\le n<2^{k_0}$. We have by (3.28) that

$$\frac{|{\sum }_{i=1}^n{X}_i|}{n^{\frac{1}{t}}}\le \underset{2^{k_0-1}\le n<2^{k_0}}{max}\frac{|{\sum }_{i=1}^n{X}_i|}{n^{\frac{1}{t}}}\le \frac{2^{\frac{2}{t}}{max}_{1\le j\le 2^{k_0}}|{\sum }_{i=1}^j{X}_i|}{2^{\frac{k_0+1}{t}}}\to 0\text{a.s., }{k}_0\to \mathrm{\infty },$$

which implies (3.26).

• (ii) Similar to the proof of (i), we can get (ii) immediately. The details are omitted. This completes the proof of the corollary. □

Remark 3.1 We point out that the cases $1+\alpha +\beta >0$ and $1+\alpha +\beta <0$ are considered in Theorem 3.1 and the case $1+\alpha +\beta =0$ is considered in Theorem 3.2, respectively. Theorem 3.1 and Theorem 3.2 consider the complete convergence for weighted sums of arrays of rowwise $\tilde{\rho }$-mixing random variables, while Theorem 3.3 considers the complete convergence for weighted sums of sequences of $\tilde{\rho }$-mixing random variables. In addition, Theorem 3.1 and Theorem 3.2 could be applied to obtain the Baum and Katz type result for arrays of rowwise $\tilde{\rho }$-mixing random variables, while Theorem 3.3 could be applied to establish the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of $\tilde{\rho }$-mixing random variables.