Complete convergence and complete moment convergence for ...

Wu et al. Journal of Inequalities and Applications (2015) 2015:200 DOI 10.1186/s13660-015-0717-1

RESEARCH

Open Access

Complete convergence and complete moment convergence for weighted sums of m-NA random variables Yongfeng Wu1* , Tien-Chung Hu2 and Andrei Volodin3 *

Correspondence: [email protected] College of Mathematics and Computer Science, Tongling University, Tongling, 244000, China Full list of author information is available at the end of the article 1

Abstract The authors study the complete convergence and the complete moment convergence for weighted sums of m-negatively associated (m-NA) random variables and obtain some new results. These results extend and improve the corresponding theorems of Sung (Stat. Pap. 52:447-454, 2011). In addition, we point out that an open problem presented in Sung (Stat. Pap. 54:773-781, 2013) can be solved by means of the method used in this paper. MSC: 60F15; 62G05 Keywords: complete convergence; complete moment convergence; weighted sums; m-negatively associated random variable

1 Introduction Let {X, Xn , n ≥ } be a sequence of random variables and {ani ,  ≤ i ≤ n, n ≥ } be an array of constants. Because the weighted sums ni= ani Xi play important roles in some useful linear statistics, many authors studied the strong convergence for the weighted sums. We refer the reader to Cuzick [], Wu [], Bai and Cheng [], Sung [], Chen and Gan [], Cai [], Wu [], Zarei and Jabbari [], Sung [], Sung [], Shen [], Chen and Sung []. The concept of the complete convergence was introduced by Hsu and Robbins []. A sequence of random variables {Un , n ≥ } is said to converge completely to a constant θ if ∞ P |Un – θ | > ε < ∞

for all ε > .

n=

Chow [] presented the following more general concept of the complete moment convergence. Let {Zn , n ≥ } be a sequence of random variables and an > , bn > , q > . If ∞

q an E b– n |Zn | – ε + < ∞ for some or all ε > ,

n=

then the above result was called the complete moment convergence. The following concept was introduced by Joag-Dev and Proschan []. © 2015 Wu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Wu et al. Journal of Inequalities and Applications (2015) 2015:200

Page 2 of 14

Definition . A finite family of random variables {Xk ,  ≤ k ≤ n} is said to be negatively associated (abbreviated to NA) if for any disjoint subsets A and B of {, , . . . , n} and any real coordinate-wise nondecreasing functions f on RA and g on RB , Cov f (Xi , i ∈ A), g(Yj , j ∈ B) ≤  whenever the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA. Definition . Let m ≥  be a fixed integer. A sequence of random variables {Xn , n ≥ } is said to be m-negatively associated (abbreviated to m-NA) if for any n ≥  and any i , . . . , in such that |ik – ij | ≥ m for all  ≤ k = j ≤ n, Xi , . . . , Xin are NA. The concept of m-NA random variables was introduced by Hu et al. []. It is easily seen that this concept is a natural extension from NA random variables (wherein m = ). It is well known that the properties of NA random variables have been applied to the reliability theory, multivariate statistical analysis and percolation theory. Sequences of NA random variables have been an attractive research topic in the recent literature. For example, Matula [], Su et al. [], Shao [], Gan and Chen [], Fu and Zhang [], Baek et al. [], Chen et al. [], Cai [], Xing [], Sung [], Qin and Li [], Wu []. Since NA implies m-NA, it is very significant to study the convergence properties of this wider mNA class. However, to the best of our knowledge, besides Hu et al. [] and Hu et al. [], few authors discuss the convergence properties for sequences of m-NA random variables. Cai [] studied the complete convergence for weighted sums of identically distributed NA random variables. He obtained the following theorem. Theorem A Let {X, Xn , n ≥ } be a sequence of identically distributed NA random variables, and let {ani ,  ≤ i ≤ n, n ≥ } be an array of constants satisfying Aα = lim sup Aα,n < ∞, n→∞

Aαα,n

=

n

|ani |α /n

(.)

i=

for some  < α ≤ . Let bn = n/α (log n)/γ for some γ > . Furthermore, suppose that EX =  when  < α ≤ . If E exp(h|X|γ ) < ∞ for some h > , then ∞ n=

m

n– P max ani Xi > bn ε < ∞ for all ε > . ≤m≤n

(.)

i=

Sung [] improved Theorem A by replacing some much weaker moment conditions. Theorem B Let {X, Xn , n ≥ } be a sequence of identically distributed NA random variables, and let {ani ,  ≤ i ≤ n, n ≥ } be an array of constants satisfying (.) for some  < α ≤ . Let bn = n/α (log n)/γ for some γ > . Furthermore, suppose that EX =  when  < α ≤ . Then the following statements hold: (i) If α > γ , then E|X|α < ∞ implies (.). (ii) If α = γ , then E|X|α log |X| < ∞ implies (.). (iii) If α < γ , then E|X|γ < ∞ implies (.).


Page 3 of 14

The main purpose of this article is to discuss the complete convergence and the complete moment convergence for weighted sums of m-NA random variables. We shall extend Theorem B to m-NA random variables. In addition, we shall extend and improve Theorem B by obtaining a much stronger conclusion under the same conditions (see Remark .). It is worthy to point out that the open problem presented in Sung [], see Remark ., can be solved by means of the method used in this article (see Remark .). Throughout this paper, the symbol C represents positive constants whose values may change from one place to another. For a finite set A the symbol (A) denotes the number of elements in the set A.

2 Preliminaries We first recall the following concept of stochastic domination, which is a slight generalization of identical distribution. An sequence of random variables {Xn , n ≥ } is said to be stochastically dominated by a random variable X (write {Xn } ≺ X) if there exists a constant C >  such that sup P |Xn | > x ≤ CP |X| > x ,

∀x > .

n≥

The following exponential inequality for m-NA random variables can be proved by means of Theorem  in Shao [] and the proof of Lemma  in Hu et al. []. Here we omit the details. Lemma . Let {Xn , n ≥ } be a sequence of m-NA random variables with zero means and j finite second moments. Let Sj = k= Xk and Bn = nk= EXk . Then for all n ≥ m, x >  and a > , P max |Sj | ≥ x ≤ mP max |Xj | > a ≤j≤n

≤j≤n

+ m exp –

x/(ma)

mBn x . + m m Bn (xa + mBn )

(.)

Remark . Since e–x ≤ ( + x)– for x > , we get, for x >  and a > ,

exp –

xa x/(ma) x xa –x/(ma) = exp – ≤ + . m Bn mBn mBn

Noting that

mBn (xa + mBn )

x/(ma)

xa = + mBn

–x/(ma)

xa –x/(ma) ≤ + . mBn

Therefore, it follows by (.) that n xa –x/(ma) P |Xk | > a + m  + . P max |Sj | ≥ x ≤ m ≤j≤n mBn

(.)

k=

Now we present a Rosenthal-type inequality for maximum partial sums of m-NA random variables, which is the crucial tool in the proof of our main results.


Page 4 of 14

Lemma . Let {Xn , n ≥ } be a sequence of m-NA random variables with mean zero and j E|Xk |q < ∞ for every  ≤ k ≤ n. Let Sj = k= Xk ,  ≤ j ≤ n. Then for q ≥ , there exists a positive constant C depending only on q such that

E max |Sj | ≤ C q

n

≤j≤n

E|Xk | + q

n

k=

Proof Let Bn =

 k= EXk .

∞

E|Y | = q

.

(.)

k=

n

q

q/ EXk

Noting that

P |Y | ≥ x xq– dx

E|Y |q < ∞ .

(.)



By taking a = x/(mq) in (.), we have

∞

E max |Sj |q = q ≤j≤n



≤ mq

P max |Sj | ≥ x xq– dx ≤j≤n

n ∞

P |Xk | ≥ x/(mq) xq– dx



k=

∞

+ mq 

x + m qBn

–q xq– dx

=: A + B. By (.), we have A = q+ q mq+ qq B = +q/ m+q q+q/ (Bn )q/

n

∞



=

+q/

m

+q +q/

q

q k= E|Xk | .

B(q/, q/)

Letting t = x /(m qBn ), then

( + t)–q t q/– dt

n

q/ EXk

,

k=

where



t α– ( – t)β– dt =

B(α, β) = 

∞

t α– ( + t)–(α+β) dt.



Letting C = max{q+ q mq+ qq , +q/ m+q q+q/ B(q/, q/)}, we can get (.). The proof is complete. Lemma . (Wang et al. []) Let {Xn , n ≥ } be a sequence of random variables with {Xn } ≺ X. Then there exists a constant C such that, for all q >  and x > , (i) E|Xk |q I(|Xk | ≤ x) ≤ C{E|X|q I(|X| ≤ x) + xq P(|X| > x)}, (ii) E|Xk |q I(|Xk | > x) ≤ CE|X|q I(|X| > x). The following lemma is very important in the proof of our result, which improves Lemma . and Lemma . of Sung []. Lemma . Let {ani ,  ≤ i ≤ n, n ≥ } be an array of constants satisfying some α > . Let bn = n/α (log n)/γ for some γ > . Then

n

α i= |ani |

≤ n for


I =:

∞

n– b–α n

n=

i=

n

α i= |ani |

Proof From

I=

≤ ≤

n

∞

⎧ ⎪CE|X|α for α > γ , ⎨ α E|ani X| I |ani X| > bn ≤ CE|X|α log |X| for α = γ , ⎪ ⎩ for α < γ . CE|X|γ ≤ n, we have

n– (log n)–α/γ

n

n=

i=

∞

n


n=

i=

∞

n


n=

≤

Page 5 of 14

E|ani X|α I |X|α > n(log n)α/γ |ani |–α

E|ani X|α I |X|α > n(log n)α/γ

n

–

|ani |α

i=

E|ani X|α I |X| > (log n)/γ

i=

∞

n– (log n)–α/γ E|X|α I |X| > (log n)/γ

n=

=

∞

n– (log n)–α/γ

E|X|α I log m < |X|γ ≤ log(m + )

m=n

n=

=

∞

∞

m E|X|α I log m < |X|γ ≤ log(m + ) n– (log n)–α/γ .

m=

n=

Observing that m

–

n (log n)

n=

–α/γ

⎧ ⎪ ⎨C ≤ C log log m ⎪ ⎩ C(log m)–α/γ

for α > γ , for α = γ , for α < γ ,

we can get ⎧ α ⎪ for α > γ , ⎨CE|X| α I ≤ CE|X| log |X| for α = γ , ⎪ ⎩ for α < γ . CE|X|γ

The proof of Lemma . is completed. Remark . Noting that ∞ n=

n–

∞ n n P |ani X| > bn ≤ n– b–α E|ani X|α I |ani X| > bn , n i=

n=

i=

we know that Lemma . improves Lemma . and Lemma . of Sung []. In addition, the method used in this paper is novel and much simpler than that in Sung [].

3 Main result In this section, we state our main results and their proofs.


Page 6 of 14

Theorem . Let {Xn , n ≥ } be a sequence of m-NA random variables with {Xn } ≺ X, and let {ani ,  ≤ i ≤ n, n ≥ } be an array of constants satisfying (.) for some  < α ≤ . Let bn = n/α (log n)/γ for some γ > . Furthermore, suppose that EXi =  when  < α ≤ . Then the following statements hold: (i) If α > γ , then E|X|α < ∞ implies (.). (ii) If α = γ , then E|X|α log |X| < ∞ implies (.). (iii) If α < γ , then E|X|γ < ∞ implies (.). Remark . Since NA implies m-NA, Theorem . extends Theorem B. Compared with Sung [], the proof of Theorem . is different from that of Theorem . in Sung []. Corollary . Let {Xn , n ≥ } be a sequence of m-NA random variables with {Xn } ≺ X, and let {ai ,  ≤ i ≤ n} be a sequence of constants satisfying Aα = lim sup Aα,n < ∞, n→∞

Aαα,n =

n

|ai |α /n

i=

for some  < α ≤ . Let bn = n/α (log n)/γ for some γ > . Furthermore, suppose that EXi =  when  < α ≤ . Then b– n

n

a i Xi → 

a.s.

i=

By a similar argument as the proof of Corollary . in Cai [], we can prove this corollary. Here we omit the details. Theorem . Assume that the conditions of Theorem . hold, then the following statements hold: (i) If α > γ , then E|X|α < ∞ implies ∞

–

n E

b– n

n=

m α max ani Xi – ε < ∞ for all ε > . ≤m≤n i=

+

(ii) If α = γ , then E|X|α log |X| < ∞ implies (.). (iii) If α < γ , then E|X|γ < ∞ implies (.). Remark . Noting that m α n– E b– ani Xi – ε n max ≤m≤n n= i= + m

∞ ∞ – – /α dt = n P bn max ani Xi > ε + t ≤m≤n  n= i= m

∞ ∞ – – /α dt. n P bn max ani Xi > ε + t = ≤m≤n 

∞

n=

i=

Therefore, Theorem . extends and improves Theorem B.

(.)


Page 7 of 14

Proof of Theorem . Without loss of generality, we may assume that ani ≥ . For fixed n ≥ , let Yni = –bn I(ani Xi < –bn ) + ani Xi I ani |Xi | ≤ bn + bn I(ani Xi > bn ), Zni = (ani Xi + bn )I(ani Xi < –bn ) + (ani Xi – bn )I(ani Xi > bn ). Then Yni + Zni = ani Xi , and it follows by the definition of m-NA and Property  of Joag-Dev and Proschan [] that {Yni , i ≥ , n ≥ } is sequence of m-NA random variables. Then ∞ n=

m

n P max ani Xi > bn ε ≤m≤n

–

i=

≤+

∞

–

n

n=

n

∞ P ani |Xi | > bn + n– P

n=

i=

m

max Yni > bn ε ≤m≤n i=

=:  + H + H . By {Xn } ≺ X and Lemma ., we have H ≤ C

∞

n–

n=

∞ n n P ani |X| > bn ≤ C n– b–α aαni E|X|α I ani |X| > bn < ∞. n n=

i=

i=

Then we prove H < ∞. Noting that either E|X|α log |X| < ∞ for α = γ , or E|X|γ < ∞ for α < γ implies E|X|α < ∞. From (.), without loss of generality, we may assume that n α i= ani ≤ n. We first prove m EYni →  as n → ∞. L =: b– n max ≤m≤n

(.)

i=

For  < α ≤ , by Lemma . and L ≤ Cb– n

n

n

α i= ani

n ani E|X|I ani |X| ≤ bn + C P ani |X| > bn

i=

≤

Cb–α n

≤ n, we have

i=

n

aαni E|X|α I

ani |X| ≤ bn + Cb–α n

i=

n

aαni E|X|α I ani |X| > bn

i=

≤ C(log n)–α/γ E|X|α →  as n → ∞. For  < α ≤ , by EXi = , |Zni | ≤ ani |Xi |I(ani |Xi | > bn ), and Lemma ., we have m n L = b– EZni ≤ b– ani E|Xi |I ani |Xi | > bn n max n ≤m≤n i=

≤ Cb– n

n

i=

n ani E|X|I ani |X| > bn ≤ Cb–α aαni E|X|α I ani |X| > bn n

i=

≤ C(log n)

i= –α/γ

E|X| →  as n → ∞. α


Page 8 of 14

Hence (.) holds for  < α ≤ . Then, while n is sufficiently large, m max EYni ≤ bn ε/. ≤m≤n

(.)

i=

Let q > max{, γ /α}. Then by (.), the Markov inequality, and Lemma ., we have m

– H ≤ n P max (Yni – EYni ) > bn ε/ ≤m≤n n= i= q m ∞ – –q n bn E max (Yni – EYni ) ≤C ≤m≤n

∞

n=

≤C

∞

n– b–q n

n=

i=

n

q/

E|Yni |



+C

∞

n– b–q n

n=

i=

n

E|Yni |q

i=

=: H + H . Firstly, we prove H < ∞. By Lemma ., α ≤ , H ≤ C

∞

–

n

n=

≤C

∞

b– n

n

b–α n

n=

≤C

∞

ani E|X| I

α i= |ani |

≤ n, and q > γ /α, we have

n ani |X| ≤ bn + P ani |X| > bn

i=

–

n

n

q/

i=

n

aαni E|X|α I

q/ n –α α α ani |X| ≤ bn + bn ani E|X| I ani |X| > bn

i=

i=

αq q/ n– (log n)– γ E|X|α < ∞.

n=

Next we consider H . By Lemma ., we have H ≤ C

∞

n– b–q n

n=

n

∞ n q ani E|X|q I ani |X| ≤ bn + C n– P ani |X| > bn n=

i=

i=

=: H + H . Similar to the proof of H < ∞, we get directly H < ∞. Then final work is to prove H < ∞. For j ≥  and n ≥ , let Inj =  ≤ i ≤ n : n/α (j + )–/α < |ani | ≤ n/α j–/α . Then {Inj , j ≥ } are disjoint, j≥ Inj = N for all n ≥  from ni= |ani |α ≤ n, where N is the set of positive integers. Noting that for all k ≥ , we have n≥

n i=

=

∞ j=k

|ani |α =

∞ j= i∈Inj

|ani |α ≥

∞

(Inj )n(j + )– ≥

j=

(Inj )n(j + )–q/α (j + )q/α– ≥

∞

(Inj )n(j + )–

j=k ∞ j=k

(Inj )n(j + )–q/α (k + )q/α– .


Page 9 of 14

Hence for all k ≥ , we have ∞

(Inj )j–q/α ≤ C(k + )–q/α .

(.)

j=k

Then H =

∞

n

n––q/α (log n)–q/γ

n=

=

i=

∞

∞


n=

≤

∞

∞


n– (log n)–q/γ

∞

n=

(Inj )j–q/α E|X|q I |X| ≤ (log n)/γ

j=

∞


n=

×

(Inj )nq/α j–q/α E|X|q I |X| ≤ (j + )/α (log n)/γ

j=

∞

+

|ani |q E|X|q I |ani X| ≤ n/α (log n)/γ

j= i∈Inj

n=

≤

|ani |q E|X|q I |ani X| ≤ n/α (log n)/γ

∞

(Inj )j–q/α

j=

j

E|X|q I k /α (log n)/γ < |X| ≤ (k + )/α (log n)/γ

k=

=:

H∗

+ H∗∗ .

By (.) and q > γ /α ≥ γ , we have H∗

≤C

∞

n– (log n)–q/γ E|X|q I |X|γ ≤ log n

n=

=C

∞

–

n (log n)

–q/γ

n=

=C

∞

E|X|q I log(m – ) < |X|γ ≤ log m

m= ∞ E|X|q I log(m – ) < |X|γ ≤ log m n– (log n)–q/γ n=m

m=

≤C

n

∞ (log m)–q/γ E|X|q I log(m – ) < |X|γ ≤ log m m=

≤ CE|X|γ < ∞. By (.), we have H∗∗ =

∞


n=

×

∞ k=

∞ E|X|q I k /α (log n)/γ < |X| ≤ (k + )/α (log n)/γ (Inj )j–q/α j=k


≤C

∞

Page 10 of 14


n=

×

∞

(k + )–q/α E|X|q I k /α (log n)/γ < |X| ≤ (k + )/α (log n)/γ

k=

≤C

∞

–

n (log n)

–α/γ

n=

=C

∞

∞

E|X|α I k /α (log n)/γ < |X| ≤ (k + )/α (log n)/γ

k=

n– (log n)–α/γ E|X|α I |X| > (log n)/γ .

n=

Noting that we obtain the following result in the proof of Lemma ., ⎧ ⎪CE|X|α for α > γ , ⎨ – –α/γ α /γ ≤ CE|X|α log |X| for α = γ , n (log n) E|X| I |X| > (log n) ⎪ ⎩ n= CE|X|γ for α < γ .

∞

Hence we get H∗∗ < ∞ combining the assumptions of Theorem .. The proof is completed. Remark . It is easily seen that the proof of H < ∞ complements Lemma . of Sung []. In fact, in that lemma Sung only proved H < ∞ for the case α = γ . It is worthy to point out that ani =  or |ani | >  is required in Sung []. Here, we do not require the extra conditions. Remark . Sung [] proved Theorem . for the case α = γ when {Xn , n ≥ } is a sequence of ρ ∗ -mixing random variables. However, he posed an open problem, that is, whether Theorem . (i.e. Theorem . in Sung []) remains true for ρ ∗ -mixing random variables. The crucial tool of the proof of Theorem . is the Rosenthal-type inequality for maximum partial sums of m-NA random variables. For ρ ∗ -mixing random variables, the Rosenthal-type inequality for maximum partial sums also holds (see Utev and Peligrad []). Therefore, it is easy to solve the above open problem by following the method used in the proof of Theorem .. Proof of Theorem . For any given ε > , we have m α n E max ani Xi – ε ≤m≤n n= i= +

∞ m ∞ – – /α = n P bn max ani Xi > ε + t dt ≤m≤n  n= i= m m

∞ ∞ ∞ – – /α dt n P max ani Xi > bn ε + n P max ani Xi > bn t ≤ ≤m≤n ≤m≤n 

∞

–

b– n

n=

=: I + I .

i=

n=

i=


Page 11 of 14

Therefore, to prove (.), one needs only to prove that I < ∞ and I < ∞. By Theorem ., we get directly I < ∞. For all t ≥ , we denote Yni = –bn t /α I ani Xi < –bn t /α + ani Xi I ani |Xi | ≤ bn t /α + bn t /α I ani Xi > bn t /α , Zni = ani Xi – Yni . Then I ≤

∞

n–

P max ani |Xi | > bn t /α dt ≤i≤n



n=

+

∞

∞

∞

–

n



n=

m

/α P max Yni > bn t dt ≤m≤n

i=

=: I + I . Noting that

∞



α α P ani |X| > bn t /α dt ≤ b–α n ani E|X| I |ani X| > bn ,

by {Xi } ≺ X, Lemma ., and the assumptions of Theorem ., we have I ≤

∞

n–

n=

≤

∞

n 

i=

n– b–α n

∞ n P ani |Xi | > bn t /α dt ≤ n–

∞

n=

n

n=

i=

∞

P ani |X| > bn t /α dt



aαni E|X|α I ani |X| > bn < ∞.

i=

Next we prove that I < ∞. We first show

J

m = sup t –/α b– max EY → ni n ≤m≤n t≥ i=

For  < α ≤ , by Lemma . and J ≤ C sup t –/α b– n t≥

α i= ani

t≥

n

n ani E|X|I ani |X| ≤ bn t /α + C sup P ani |X| > bn t /α t≥ i=

n aαni E|X|α I ani |X| ≤ bn t /α + Cb–α aαni E|X|α n

i= –α/γ

(.)

≤ n, we have

i=

≤ C sup t – b–α n ≤ C(log n)

n

n

as n → ∞.

i=

E|X| →  as n → ∞. α

For  < α ≤ , by EXi = , |Zni | ≤ ani |Xi |I(ani |Xi | > bn t /α ), and Lemma ., we have

J =

m sup t –/α b– max EZ ni n ≤m≤n t≥ i=

≤ C sup t –/α b– n t≥

n i=

ani E|X|I ani |X| > bn t /α


n

≤ C sup t – b–α n t≥

Page 12 of 14

aαni E|X|α I ani |X| > bn t /α

i=

≤ C(log n)

–α/γ

E|X|α →  as n → ∞.

From (.), we know that, while n is sufficiently large, m max EYni ≤ bn t /α / ≤m≤n

(.)

i=

holds uniformly for t ≥ . Let q > max{, γ /α}. Then by (.) and Lemma ., we have m

/α I ≤ n P max (Yni – EYni ) > bn t / dt ≤m≤n  n= i= q ∞ ∞ m ≤C n– b–q t –q/α E max (Yni – EYni ) dt n ≤m≤n 

∞

∞

–

n=

≤C

∞

i=

n– b–q n

∞

t

n

–q/α



n=

E|Yni | dt + C q

∞

n– b–q n

n=

i=

t 

=: I + I . By Lemma ., α ≤ , and q > γ /α, we have I ≤ C

∞

n– b–q n

t

i=

–

∞

– b–α n t



– + b–α n t

n

n–

aαni E|X|α I

≤C

∞

∞



n=

n

q/ dt

aαni E|X|α I ani |X| ≤ bn t /α

i=

i=

≤C

ani E|X| I ani |X| ≤ bn t /α

i=

n

n=

∞

n

n P ani |X| > bn t /α

+ bn t /α ≤C

–q/α



n=

∞

∞

ani |X| > bn t

t –q/ b–α n

n

/α

q/ dt

q/ aαni E|X|α

dt

i=

q/ n– (log n)–αq/(γ ) E|X|α < ∞.

n=

For I , we have I ≤ C

∞

n– b–q n

n=

+C

∞ n=

n 

i=

n–

n i=

∞

∞ 

q t –q/α ani E|X|q I ani |X| ≤ bn t /α dt


∞

–q/α

n i=

q/ E|Yni |



dt


=C

∞

n– b–q n

n

n=

+C

n– b–q n

i=

∞

n–

n ∞

n=

q t –q/α ani E|X|q I ani |X| ≤ bn dt

n ∞

n=

+C



i=

∞

∞



Page 13 of 14

q t –q/α ani E|X|q I bn < ani |X| ≤ bn t /α dt




i=

=: I + I + I . Similar to the proof of I < ∞, we get I < ∞. Similar to the proof of H < ∞, we get I < ∞. By q >  ≥ α and the following standard arguments, we get b–q n

∞ 


≤ b–q n

∞

m

m=

≤ b–q n

∞

m+


q m–q/α ani E|X|q I bn < ani |X| ≤ bn (m + )/α

m=

≤ b–q n

∞

m–q/α

m=

≤ b–q n

∞

m

q ani E|X|q I bn s/α < ani |X| ≤ bn (s + )/α

s= ∞ q ani E|X|q I bn s/α < ani |X| ≤ bn (s + )/α m–q/α m=s

s=

≤ Cb–q n

∞

q s–q/α ani E|X|q I bn s/α < ani |X| ≤ bn (s + )/α

s=

≤ Cb–α n

∞

aαni E|X|α I bn s/α < ani |X| ≤ bn (s + )/α

s=

α α ≤ Cb–α n ani E|X| I ani |X| > bn . Hence by Lemma ., we have I ≤ C

∞ n=

n– b–α n

n

aαni E|X|α I ani |X| > bn < ∞.

i=

The proof is completed.

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 College of Mathematics and Computer Science, Tongling University, Tongling, 244000, China. 2 Department of Mathematics, National Tsing Hua University, Hsinchu, 300, Taiwan, ROC. 3 Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada.


Page 14 of 14

Acknowledgements The authors are grateful to the referee for carefully reading the manuscript and for offering comments which enabled them to improve the paper. The research of Y Wu was supported by the Natural Science Foundation of Anhui Province (1308085MA03, 1408085MA03), the Key Grant Project for Backup Academic Leaders of Tongling University (2014tlxyxs21) and the Key NSF of Anhui Educational Committee (KJ2014A255). Received: 23 December 2014 Accepted: 28 May 2015 References 1. Cuzick, J: A strong law for weighted sums of i.i.d. random variables. J. Theor. Probab. 8, 625-641 (1995) 2. Wu, WB: On the strong convergence of a weighted sum. Stat. Probab. Lett. 44, 19-22 (1999) 3. Bai, ZD, Cheng, PE: Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46, 105-112 (2000) 4. Sung, SH: Strong laws for weighted sums of i.i.d. random variables. Stat. Probab. Lett. 52, 413-419 (2001) 5. Chen, PY, Gan, SX: Limiting behavior of weighted sums of i.i.d. random variables. Stat. Probab. Lett. 77, 1589-1599 (2007) 6. Cai, GH: Strong laws for weighted sums of NA random variables. Metrika 68, 323-331 (2008) 7. Wu, QY: Complete convergence for weighted sums of sequences of negatively dependent random variables. J. Probab. Stat. 2011, Article ID 202015 (2011) 8. Zarei, H, Jabbari, H: Complete convergence of weighted sums under negative dependence. Stat. Pap. 52, 413-418 (2011) 9. Sung, SH: On the strong convergence for weighted sums of ρ ∗ -mixing random variables. Stat. Pap. 54, 773-781 (2013) 10. Sung, SH: On the strong convergence for weighted sums of random variables. Stat. Pap. 52, 447-454 (2011) 11. Shen, AT: On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. (2012). doi:10.1007/s13398-012-0067-5 12. Chen, PY, Sung, SH: On the strong convergence for weighted sums of negatively associated random variables. Stat. Probab. Lett. 92, 45-52 (2014) 13. Hsu, PL, Robbins, H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. 33, 25-31 (1947) 14. Chow, YS: On the rate of moment complete convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16, 177-201 (1988) 15. Joag-Dev, K, Proschan, F: Negative association of random variables with applications. Ann. Stat. 11, 286-295 (1983) 16. Hu, YJ, Ming, RX, Yang, WQ: Large deviations and moderate deviations for m-negatively associated random variables. Acta Math. Sci. 27, 886-896 (2007) 17. Matula, P: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 15, 209-213 (1992) 18. Su, C, Zhao, LC, Wang, YB: Moment inequalities and weak convergence for negatively associated sequences. Sci. China Ser. A 40, 172-182 (1997) 19. Shao, QM: A comparison theorem on maximum inequalities between negatively associated and independent random variables. J. Theor. Probab. 13, 343-356 (2000) 20. Gan, SX, Chen, PY: On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables. Acta Math. Sci. 27, 283-290 (2007) 21. Fu, KA, Zhang, LX: Precise rates in the law of the logarithm for negatively associated random variables. Comput. Math. Appl. 54, 687-698 (2007) 22. Baek, JI, Choi, IB, Niu, SL: On the complete convergence of weighted sums for arrays of negatively associated variables. J. Korean Stat. Soc. 37, 73-80 (2008) 23. Chen, PY, Hu, TC, Liu, X, Volodin, A: On complete convergence for arrays of rowwise negatively associated random variables. Theory Probab. Appl. 52, 323-328 (2008) 24. Xing, GD: On the exponential inequalities for strictly stationary and negatively associated random variables. J. Stat. Plan. Inference 139, 3453-3460 (2009) 25. Qin, YS, Li, YH: Empirical likelihood for linear models under negatively associated errors. J. Multivar. Anal. 102, 153-163 (2011) 26. Wu, YF: Convergence properties of the maximal partial sums for arrays of rowwise NA random variables. Theory Probab. Appl. 56, 527-535 (2012) 27. Hu, TC, Chiang, CY, Taylor, RL: On complete convergence for arrays of rowwise m-negatively associated random variables. Nonlinear Anal., Theory Methods Appl. 71, e1075-e1081 (2009) 28. Wang, XJ, Li, XQ, Yang, WZ, Hu, SH: On complete convergence for arrays of rowwise weakly dependent random variables. Appl. Math. Lett. 25, 1916-1920 (2012) 29. Utev, S, Peligrad, M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theor. Probab. 16, 101-115 (2003)