Complete convergence and complete moment convergence for arrays of rowwise ANA random variables

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R E S E A R C H

Open Access

Complete convergence and complete

moment convergence for arrays of rowwise

ANA random variables

Haiwu Huang

1,2*

_{, Jiangyan Peng}

3

_{, Xiongtao Wu}

1

_{and Bin Wang}

2

*_{Correspondence:}

haiwuhuang@126.com

1_{College of Mathematics and}

Statistics, Hengyang Normal University, Hengyang, 421002, P.R. China

2_{College of Science, Guilin}

University of Technology, Guilin, 541004, P.R. China

Full list of author information is available at the end of the article

Abstract

In this article, we investigate complete convergence and complete moment convergence for weighted sums of arrays of rowwise asymptotically negatively associated (ANA) random variables. The results obtained not only generalize the corresponding ones of Sung (Stat. Pap. 52:447-454, 2011), Zhouet al.(J. Inequal. Appl. 2011:157816, 2011), and Sung (Stat. Pap. 54:773-781, 2013) to the case of ANA random variables, but also improve them.

MSC: 60F15

Keywords: arrays of rowwise ANA random variables; complete convergence; complete moment convergence; weighted sums

1 Introduction

Recently, Sung [] proved the following strong laws of large numbers for weighted sums of identically distributed negatively associated (NA) random variables.

Theorem A Let{Xn,n≥}be a sequence of identically distributed NA random variables,

and let{ani, ≤i≤n,n≥}be an array of real constants satisfying n

i=

|ani|α=O(n) (.)

for some <α≤.Let bn=n/α(logn)/γfor someγ > .Furthermore,suppose that EX= 

for <α≤.If

E|X|α<∞ forα>γ,

E|X|αlog

 +|X|

<∞ forα=γ,

E|X|γ <∞ forα<γ,

(.)

then

∞

n=

 nP

max ≤j≤n

j

i=

aniXi

>εbn

<∞ for allε> . (.)

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Zhouet al.[] partially extended Theorem A for NA random variables to the case of ˜

ρ-mixing (orρ∗-mixing) random variables by using a diﬀerent method.

Theorem B Let{Xn,n≥}be a sequence of identically distributedρ˜-mixing random

vari-ables,and let{ani, ≤i≤n,n≥}be an array of real constants satisfying n

i=

|ani|max{α,γ}=O(n) (.)

for some <α≤andγ > withα=γ.Let bn=n/α(logn)/γ.Assume that EX= for

 <α≤.If(.)is satisﬁed forα=γ,then(.)holds.

Zhouet al.[] left an open problem whether the caseα=γ of Theorem A holds for ˜

ρ-mixing random variables. Sung [] solved this problem and obtained the following re-sult.

Theorem C Let{Xn,n≥}be a sequence of identically distributedρ˜-mixing random

vari-ables,and let{ani, ≤i≤n,n≥}be an array of real constants satisfying(.)for some

 <α≤andγ > withα=γ.Let bn=n/α(logn)/α.Assume that EX= for <α≤.

If(.)is satisﬁed forα=γ,then(.)holds.

Inspired by these theorems, in this paper, we further investigate the limit convergence properties and obtain some much stronger conclusions, which extend and improve The-orems A, B, and C to a wider class of dependent random variables under the same condi-tions.

Now we introduce some deﬁnitions of dependent structures.

Deﬁnition . A ﬁnite family of random variables{Xi, ≤i≤n}is said to be NA if for

any disjoint subsetsAandBof{, , . . . ,n},

Covf(Xi,i∈A),f(Xj,j∈B)

≤ (.)

whenever f andf are any real coordinatewise nondecreasing functions such that this

covariance exists. An inﬁnite family of random variables{Xn,n≥}is NA if every ﬁnite

its subfamily is NA.

Deﬁnition . A sequence of random variables{Xn,n≥}is calledρ˜-mixing if for some

integern≥, the mixing coeﬃcient

˜

ρ(s) =supρ(S,T) :S,T⊂N,dist(S,T)≥s → ass→ ∞, (.)

where

ρ(S,T) =sup

|EXY–EXEY| √

VarX·√VarY;X∈L

σ(S),Y∈L

σ(T), (.)

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Deﬁnition . A sequence of random variables{Xn,n≥}is said to be ANA if

ρ–(s) =supρ–(S,T) :S,T⊂N,dist(S,T)≥s → ass→ ∞, (.)

where

ρ–(S,T) = ∨

_Cov

(f(Xi,i∈S),g(Xj,j∈T))

(Varf(Xi,i∈S))/(Varg(Xj,j∈T))/

,f,g∈C

, (.)

andCis the set of nondecreasing functions.

An array of random variables{Xni,i≥,n≥}is said to be rowwise ANA random

vari-ables if for everyn≥,{Xni,i≥}is a sequence of ANA random variables.

It is obvious to see thatρ–(s)≤ ˜ρ(s) and that a sequence of ANA random variables is NA if and only ifρ–() = . So, ANA random variables includeρ˜-mixing and NA random variables. Consequently, the study of the limit convergence properties for ANA random variables is of much interest. Since the concept of ANA random variables was introduced by Zhang and Wang [], many applications have been found. For example, Zhang and Wang [] and Zhang [, ] obtained moment inequalities for partial sums, the central limit theorems, the complete convergence, and the strong law of large numbers, Wang and Lu [] established some inequalities for the maximum of partial sums and weak con-vergence, Wang and Zhang [] obtained the law of the iterated logarithm, Liu and Liu [] showed moments of the maximum of normed partial sums, Yuan and Wu [] obtained the limiting behavior of the maximum of partial sums, Budsaba et al.[] investigated the complete convergence for moving-average process based on a sequence of ANA and NA random variables, Tanet al.[] obtained the almost sure central limit theorem, Ko [] obtained the Hájek-Rényi inequality and the strong law of large numbers, Zhang [] established the complete moment convergence for moving-average process generated by ANA random variables, and so forth.

In this work, we further study the strong convergence for weighted sums of arrays of ANA random variables without assumption of identical distribution and obtain some im-proved results (i.e., the so-called complete moment convergence, which will be introduced in Deﬁnition .). As applications, the complete convergence theorems for weighted sums of arrays of identically distributed NA andρ˜-mixing random variables can been obtained. The results obtained not only generalize the corresponding ones of Sung [, ] and Zhou et al.[], but also improve them under the same conditions.

For the proofs of the main results, we need to restate some deﬁnitions used in this pa-per.

Deﬁnition . A sequence of random variables{Xn,n≥}converges completely to a

con-stantλif for allε> ,

∞

n=

P|Xn–λ|>ε

<∞.

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Deﬁnition .(Chow []) Let{Xn,n≥}be a sequence of random variables, andan> ,

bn> ,q> . If for allε≥,

∞

n=

anE

b–_n |Xn|–ε

q

+<∞,

then the sequence{Xn,n≥}is said to satisfy the complete moment convergence.

Deﬁnition . A sequence of random variables{Xn,n≥} is said to be stochastically

dominated by a random variableXif there exists a positive constantCsuch that

P|Xn| ≥x

≤CP|X| ≥x

for allx≥ andn≥.

An array of rowwise random variables{Xni,i≥,n≥}is said to be stochastically

dom-inated by a random variableXif there exists a positive constantCsuch that

P|Xni| ≥x

≤CP|X| ≥x

for allx≥,i≥ andn≥.

Throughout this paper,I(A) is the indicator function of a setA. The symbolCdenotes a positive constant, which may be diﬀerent in various places, andan=O(bn) means that

an≤Cbn.

2 Main results

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

Theorem . Let{Xni,i≥,n≥}be an array of rowwise ANA random variables

stochas-tically dominated by a random variable X,let{ani, ≤i≤n,n≥}be an array of constants

satisfying(.)for some <α≤,and let bn=n/α(logn)/γ for someγ > .Furthermore,

assume that EXni= for <α≤.If

E|X|α<∞ forα>γ,

E|X|αlog +|X|<∞ forα=γ,

E|X|γ_<_∞ _for_α_<_γ_,

(.)

then

∞

n=

 nP

max ≤j≤n

j

i=

aniXni

>εbn

<∞ for allε> . (.)

Theorem . Let{Xni,i≥,n≥}be an array of rowwise ANA random variables which

is stochastically dominated by a random variable X,let{ani, ≤i≤n,n≥}be an array of

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Furthermore,assume that EXni= for <α≤.If(.)holds,then,for <q<α,

∞

n=

 nE

 bn

max ≤j≤n

j

i=

aniXni

–ε

q

+

<∞ for allε> . (.)

Remark . Since ANA includes NA andρ˜-mixing, Theorem . extends Theorem A for identically distributed NA random variables and Theorems B and C for identically distributedρ˜-mixing random variables (by takingXni=Xiin Theorem .).

Remark . Under the conditions of Theorem ., we can obtain that

∞>

∞

n=

 nE

 bn

max ≤j≤n

j

i=

aniXni

–ε

q

+

=

∞

n=

 n

∞



P

 bn

max ≤j≤n

j

i=

aniXni

–ε>t/q

dt

≥C





∞

n=

 nP

max ≤j≤n

j

i=

aniXni

>bnε

dt

=C

∞

n=

 nP

max ≤j≤n

j

i=

aniXni

>bnε

for allε> . (.)

Hence, from (.) we get that the complete moment convergence implies the complete convergence. Compared with the results of Sung [, ] and Zhouet al.[], it is worth point-ing out that our main result is much stronger under the same conditions. So, Theorem . is an extension and improvement of the corresponding ones of Sung [, ] and Zhouet al.[].

3 Proofs

To prove the main results, we need the following lemmas.

Lemma .(Wang and Lu []) Let{Xn,n≥}be a sequence of ANA random variables.If

{fn,n≥}is a sequence of real nondecreasing(or nonincreasing)functions,then{fn(Xn),n≥

}is still a sequence of ANA random variables.

From Wang and Lu’s [] Rosenthal-type inequality for ANA random variables we obtain the following result.

Lemma .(Wang and Lu []) For a positive integerN≥and≤s< _,let{Xn,n≥}

be a sequence of ANA random variables withρ–_(N)_≤_s,_EX

n= ,and E|Xn|<∞.Then

for all n≥,there exists a positive constant C=C(, N,s)such that

E

max ≤j≤n

j

i=

Xi



≤C

n

i=

EX_i. (.)

Lemma .(Adler and Rosalsky []; Adleret al.[]) Suppose that{Xni,i≥,n≥}is

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all q> and x> ,

E|Xni|qI

|Xni| ≤x

≤CE|X|qI|X| ≤x+xqP|X|>x, (.) E|Xni|qI

|Xni|>x

≤CE|X|qI|X|>x. (.)

Lemma .(Wuet al.[]) Let{ani,i≥,n≥}be an array of real constants satisfying

(.)for someα> ,and X be a random variable.Let bn=n/α(logn)/γ for someγ > .If

p>max{α,γ},then

∞

n=

 nbpn

n

i=

E|aniX|pI

|aniX| ≤bn

≤

⎧ ⎪ ⎨ ⎪ ⎩

CE|X|α _for_α_>_γ_,

CE|X|α_log_{( +}_|_X_|₎ _for_α₌_γ_,

CE|X|γ _for_α_<_γ_.

(.)

Proof of Theorem. Without loss of generality, assume thatani≥ (otherwise, we shall

usea+_nianda–_niinstead ofani, and note thatani=a+ni–a–ni). For ﬁxedn≥, deﬁne

Yni= –bnI(aniXni< –bn) +aniXniI

|aniXni| ≤bn

+bnI(aniXni>bn), i≥;

Zni=aniXni–Yni= (aniXni+bn)I(aniXni< –bn) + (aniXni–bn)I(aniXni>bn);

A=

n

i=

(Yni=aniXni), B=A¯= n

i=

(Yni=aniXni) = n

i=

|aniXni|>bn

;

Eni=

max ≤j≤n

j

i=

aniXni

>εbn

.

It is easy to check that for allε> ,

Eni=EniA∪EniB⊂

max ≤j≤n

j

i=

Yni

>εbn

∪

_n

i=

|aniXni|>bn

,

which implies that

P(Eni)≤P

max ≤j≤n

j

i=

Yni

>εbn

+P

_n

i=

|aniXni|>bn

≤P

max ≤j≤n

j

i=

(Yni–EYni)

>εbn–max

≤j≤n

j

i=

EYni

+

n

i=

P|aniXni|>bn

. (.)

First, we shall show that

 bn

max ≤j≤n

j

i=

EYni

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For  <α≤, it follows from (.) of Lemma ., the Markov inequality, andE|X|α<∞ that

 bn

max ≤j≤n

j

i=

EYni

≤C

 bn

n

i=

|EYni|

≤C  bn

n

i=

E|aniXni|I

|aniXni| ≤bn

+C

n

i=

P|aniXni|>bn

≤C  bn

n

i=

E|aniX|I

|aniX| ≤bn

+bnP

|aniX|>bn

+C

n

i=

P|aniX|>bn

≤C  bα

n n

i=

aα

niE|X|αI

|aniX| ≤bn

+C  bα

n n

i=

aα

niE|X|α

≤C(logn)–α/γE|X|α_→_ _as_n_{→ ∞}_. _(.)

From the deﬁnition of Zni =aniXni –Yni we know that when aniXni >bn,  <Zni =

aniXni–bn<aniXni, and whenaniXni< –bn,aniXni<Zni=aniXni+bn< . Hence,|Zni|<

|aniXni|I(|aniXni|>bn).

For  <α≤, it follows fromEXni= , (.) of Lemma ., andE|X|α<∞again that

 bn

max ≤j≤n

j

i=

EYni

=

 bn

max ≤j≤n

j

i=

EZni

≤C  bn

n

i=

E|Zni|

≤C  bn

n

i=

E|aniXni|I

|aniXni|>bn

≤C  bn

n

i=

E|aniX|I

|aniX|>bn

≤C  bα

n n

i=

aα_niE|X|αI|aniX|>bn

≤C(logn)–α/γE|X|α→ asn→ ∞. (.)

By (.) and (.) we immediately obtain (.). Hence, fornlarge enough,

P(En)≤P

max ≤j≤n

j

i=

(Yni–EYni)

>

εbn



+

n

i=

P|aniXni|>bn

. (.)

To prove (.), it suﬃces to show that

I

∞

n=

 nP

max ≤j≤n

j

i=

(Yni–EYni)

>

εbn



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J

∞

n=

 n

n

i=

P|aniXni|>bn

<∞. (.)

By Lemma . it obviously follows that{Yni–EYni,i≥,n≥}is still an array of rowwise

ANA random variables. Hence, it follows from the Markov inequality and Lemma . that

I≤C

∞

n=

 n

 b

n

E

max ≤j≤n

j

i=

(Yni–EYni)



≤C

∞

n=

 n

 b

n n

i=

E|Yni–EYni|

≤C

∞

n=

 n

 b

n n

i=

EY_ni

≤C

∞

n=

 n

 b

n n

i=

E|aniXni|I

|aniXni| ≤bn

+C

∞

n=

 n

n

i=

P|aniXni|>bn

≤C

∞

n=

 n

 b

n n

i=

E|aniX|I

|aniX| ≤bn

+C

∞

n=

 n

 bα

n n

i=

E|aniX|αI

|aniX|>bn

I+I. (.)

From Lemma . (forp= ) and (.) we obtain thatI<∞. Hence, it follows from (.)

of Lemma . and from (.) that

I =C

∞

n=

 n(logn)

–α/γ

n

i=

E|aniX|αI

|aniX|α>n(logn)α/γ

≤C

∞

n=

 n(logn)

–α/γ

n

i=

E|aniX|αI

|X|α_>n(logn) α/γ

n i=|ani|α

≤C

∞

n=

 n(logn)

–α/γ

n

i=

E|aniX|αI

|X|> (logn)/γ

≤C

∞

n=

 n(logn)

–α/γ

E|X|α_I_|_X_|_{> (}_log_n)/γ

=C

∞

n=

 n(logn)

–α/γ

∞

k=n

E|X|αI(logk)/γ <|X|<log(k+ )/γ

=C

∞

k=

E|X|αI(logk)/γ<|X|<log(k+ )/γ

k

n=

 n(logn)

–α/γ

.

Note that

k

n=

 n(logn)

–α/γ

=

⎧ ⎪ ⎨ ⎪ ⎩

C forα>γ,

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Therefore, we obtain that

I≤ ⎧ ⎪ ⎨ ⎪ ⎩

CE|X|α forα>γ, CE|X|αlog( +|X|) forα=γ, CE|X|γ _for_α_<_γ_.

Under the conditions of Theorem . it follows thatI<∞.

By (.) of Lemma . and the proof ofI<∞,

J≤ ∞ n=  n  bα n n i=

E|aniX|αI

|aniX|>bn

<∞. (.)

The proof of Theorem . is completed.

Proof of Theorem. For allε> , we have

∞ n=  nE  bn max ≤j≤n

j i=

aniXni

–ε q + = ∞ n=  n ∞  P  bn max ≤j≤n

j i=

aniXni

–ε>t/q dt = ∞ n=  n   P  bn max ≤j≤n

j i=

aniXni

>ε+t/q dt + ∞ n=  n ∞  P  bn max ≤j≤n

j i=

aniXni

>ε+t/q dt ≤ ∞ n=  nP max ≤j≤n

j i=

aniXni

>εbn

+ ∞ n=  n ∞  P max ≤j≤n

j i=

aniXni

>bnt/q

dt

K+K. (.)

To prove (.), it suﬃces to proveK<∞andK<∞. By Theorem . we obtain that

K<∞. By applying a similar notation and the methods of Theorem ., for ﬁxedn≥,

i≥, and allt≥, deﬁne

Y_ni = –bnt/qI

aniXni< –bnt/q

+aniXniI

|aniXni| ≤bnt/q

+bnt/qI

aniXni>bnt/q

;

Z_ni=aniXni–Yni

=aniXni+bnt/q

IaniXni< –bnt/q

+aniXni–bnt/q

IaniXni>bnt/q

;

A=

n

i=

Y_ni =aniXni

, B=A¯=

n

i=

Y_ni =aniXni

= n i=

|aniXni|>bnt/q

;

E_ni=

max ≤j≤n

j i=

aniXni

>bnt/q

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It is easy to check that for allε> ,

PE_ni≤P

max ≤j≤n

j

i=

Y_ni

>bnt/q

+P

_n

i=

≤P

max ≤j≤n

j

i=

Y_ni –EY_ni

>bnt/q–max

≤j≤n

j

i=

EY_ni

+

n

i=

P|aniXni|>bnt/q

. (.)

First, we shall show that

max

t≥

 bnt/q

max ≤j≤n

j

i=

EY_ni

→ asn→ ∞. (.)

Similarly to the proofs of (.) and (.), for  <α≤, it follows from (.) of Lemma ., the Markov inequality, andE|X|α_<_∞_that

max

t≥

 bnt/q

max ≤j≤n

j

i=

EY_ni

≤Cmaxt≥

 bnt/q

n

i= EY_ni

≤Cmax

t≥

 bnt/q

n

i=

E|aniXni|I

+Cmax

t≥

n

i=

≤Cmax

t≥

 bnt/q

n

i=

E|aniX|I

|aniX| ≤bnt/q

+Cmax

t≥

n

i=

P|aniX|>bnt/q

≤Cmax

t≥

 bα

ntα/q n

i=

aα_niE|X|αI|aniX| ≤bnt/q

+Cmax

t≥

 bα

ntα/q n

i=

aα

niE|X|α

≤C(logn)–α/γE|X|α→ asn→ ∞. (.)

Noting that|Z_ni |<|aniXni|I(|aniXni|>bnt/q), for  <α≤, it follows fromEXn= , (.)

of Lemma ., andE|X|α<∞again that

max

t≥

 bnt/q

max ≤j≤n

j

i=

EY_ni

=maxt≥

 bnt/q

max ≤j≤n

j

i=

EZ_ni

≤Cmax

t≥

 bnt/q

n

i=

(11)

≤Cmax

t≥

 bnt/q

n

i=

E|aniXni|I

≤Cmax

t≥

 bnt/q

n

i=

E|aniX|I

|aniX|>bnt/q

≤Cmax

t≥

 bα

ntα/q n

i=

aα_niE|X|αI|aniX|>bnt/q

≤C(logn)–α/γE|X|α→ asn→ ∞. (.)

To proveK<∞, it suﬃces to show that

K ∞ n=  n ∞  P max ≤j≤n

j i=

Y_ni –EY_ni

>

bnt/q



dt<∞, (.)

K ∞ n=  n ∞  n i=

dt<∞. (.)

Hence, it follows from the Markov inequality and Lemma . that

K≤C

∞ n=  n ∞   b

nt/q

E

max ≤j≤n

j i=

Y_ni –EY_ni

 dt ≤C ∞ n=  n ∞   b

nt/q n

i=

EY_ni –EY_nidt

≤C ∞ n=  n _∞  n i=

dt +C ∞ n=  nb n ∞   t/q

n

i=

E|aniXni|I

dt ≤C ∞ n=  n ∞  n i=

dt +C ∞ n=  nb n ∞   t/q

n

i=

E|aniX|I

|aniX| ≤bn

dt +C ∞ n=  nb n ∞   t/q

n

i=

E|aniX|I

bn<|aniX| ≤bnt/q

dt. (.)

For  <q<αandn_i_=|ani|α=O(n), similarly as in the proof ofI<∞, we obtain that

K≤

∞ n=  n ∞  n i=

dt ≤C ∞ n=  n ∞  n i=

dt ≤C ∞ n=  n _∞  n i= P _|

aniX|q

bqn

>t

(12)

≤C ∞ n=  n n i=

E|aniX|q

bqn

I|aniX|>bn

≤C ∞ n=  nbα n n i=

E|aniX|αI

|aniX|>bn

<∞ (see the proof ofI<∞).

For  <q<α≤, it follows from Lemma . and (.) that

∇ ∞ n=  nb n ∞   t/q

n

i=

E|aniX|I

|aniX| ≤bn

dt ≤C ∞ n=  n  b n n i=

E|aniX|I

|aniX| ≤bn

<∞.

Takingt=xq, by the Markov inequality from (.) of Lemma . it follows that

∇ ∞ n=  nb n ∞   t/q

n

i=

E|aniX|I

bn<|aniX| ≤bnt/q

dt ≤C ∞ n=  nb n ∞ 

xq–

n

i=

E|aniX|I

bn<|aniX| ≤bnx

dx ≤C ∞ n=  nb n ∞ m= m+ m

xq–

n

i=

E|aniX|I

bn<|aniX| ≤bnx

dx ≤C ∞ n=  nb n ∞ m=

mq–

n

i=

E|aniX|I

bn<|aniX| ≤bn(m+ )

=C ∞ n=  nb n n i= ∞ m= m s=

mq–E|aniX|I

bns<|aniX| ≤bn(s+ )

=C ∞ n=  nb n n i= ∞ s=

E|aniX|I

∞

m=s

mq–

≤C ∞ n=  nb n n i= ∞ s=

E|aniX|I

sq–

≤C

∞

n=

 nbqn

n

i=

E|aniX|qI

|aniX|>bn

≤C ∞ n=  nbα n n i=

E|aniX|αI

|aniX|>bn

<∞ (see the proof ofI<∞).

The proof of Theorem . is completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

(13)

Author details

1_{College of Mathematics and Statistics, Hengyang Normal University, Hengyang, 421002, P.R. China.}2_{College of Science,}

Guilin University of Technology, Guilin, 541004, P.R. China.3_{School of Mathematical Science, University of Electronic}

Science and Technology of China, Chengdu, 610054, P.R. China.

Authors’ information

Haiwu Huang is an associate Professor, Doctor, working in the ﬁeld of probability and statistics.

Acknowledgements

The authors are most grateful to the Editor prof. A Volodin and the anonymous referees for carefully reading the paper and for oﬀering valuable suggestions, which greatly improved this paper. This paper is supported by the National Nature Science Foundation of China (11526085, 71501025, 11401127), the Humanities and Social Sciences Foundation for the Youth Scholars of Ministry of Education of China (15YJCZH066), the Guangxi Provincial Natural Science Foundation of China (2014GXNSFBA118006, 2014GXNSFCA118015), the Scientiﬁc Research Fund of Guangxi Provincial Education Department (2013YB104), the Construct Program of the Key Discipline in Hunan Province (No. [2011]76), and the Science Foundation of Hengyang Normal University (14B30).

Received: 30 August 2015 Accepted: 8 February 2016

References

1. Sung, SH: On the strong convergence for weighted sums of random variables. Stat. Pap.52, 447-454 (2011) 2. Zhou, XC, Tan, CC, Lin, JG: On the strong laws for weighted sums ofρ∗-mixing random variables. J. Inequal. Appl.

2011, Article ID 157816 (2011)

3. Sung, SH: On the strong convergence for weighted sums ofρ∗-mixing random variables. Stat. Pap.54, 773-781 (2013)

4. Zhang, LX, Wang, XY: Convergence rates in the strong laws of asymptotically negatively associated random ﬁelds. Appl. Math. J. Chin. Univ. Ser. B14(4), 406-416 (1999)

5. Zhang, LX: A functional central limit theorem for asymptotically negatively dependent random ﬁelds. Acta Math. Hung.86(3), 237-259 (2000)

6. Zhang, LX: Central limit theorems for asymptotically negatively associated random ﬁelds. Acta Math. Sin. Engl. Ser. 16(4), 691-710 (2000)

7. Wang, JF, Lu, FB: Inequalities of maximum partial sums and weak convergence for a class of weak dependent random variables. Acta Math. Sin. Engl. Ser.22(3), 693-700 (2006)

8. Wang, JF, Zhang, LX: A Berry-Esseen theorem and a law of the iterated logarithm for asymptotically negatively associated sequences. Acta Math. Sin. Engl. Ser.23(1), 127-136 (2007)

9. Liu, XD, Liu, JX: Moments of the maximum of normed partial sums ofρ–_{-mixing random variables. Appl. Math.}

J. Chin. Univ. Ser. B24(3), 355-360 (2009)

10. Yuan, DM, Wu, XS: Limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesàro alpha-integrability assumption. J. Stat. Plan. Inference140, 2395-2402 (2010) 11. Budsaba, K, Chen, PY, Volodin, A: Limiting behavior of moving average processes based on a sequence ofρ–_-mixing

and NA random variables. Lobachevskii J. Math.26, 17-25 (2007)

12. Tan, XL, Zhang, Y, Zhang, Y: An almost sure central limit theorem of products of partial sums forρ–_-mixing

sequences. J. Inequal. Appl.2012, 51 (2012). doi:10.1186/1029-242X-2012-51

13. Ko, MH: The Hájek-Rényi inequality and strong law of large numbers for ANA random variables. J. Inequal. Appl.2014, 521 (2014). doi:10.1186/1029-242X-2014-521

14. Zhang, Y: Complete moment convergence for moving average process generated byρ–_{-mixing random variables.}

J. Inequal. Appl.2015, 245 (2015). doi:10.1186/s13660-015-0766-5

15. Hsu, PL, Robbins, H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA33(2), 25-31 (1947)

16. Chow, YS: On the rate of moment complete convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16, 177-201 (1988)

17. Adler, A, Rosalsky, A: Some general strong laws for weighted sums of stochastically dominated random variables. Stoch. Anal. Appl.5, 1-16 (1987)

18. Adler, A, Rosalsky, A, Taylor, RL: Strong laws of large numbers for weighted sums of random variables in normed linear spaces. Int. J. Math. Math. Sci.12, 507-530 (1989)