# Strong convergence properties for ψ mixing random variables

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## random variables

1

### and Ling Tang

2*

*Correspondence:

2Department of Mathematics and

Physics, Anhui Jianzhu University, Hefei, 230022, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, by using the Rosenthal-type maximal inequality for

### ψ

-mixing random variables, we obtain the Khintchine-Kolmogorov-type convergence theorem, which can be applied to establish the three series theorem and the Chung-type strong law of large numbers for

### ψ

-mixing random variables. In addition, the strong stability for weighted sums of

### ψ

-mixing random variables is studied, which generalizes the corresponding one of independent random variables.

MSC: 60F15

Keywords: strong stability; Khintchine-Kolmogorov-type convergence theorem;

### ψ

-mixing random variables

1 Introduction

Let (,F,P) be a ﬁxed probability space. The random variables we deal with are all deﬁned on (,F,P). Throughout the paper, letI(A) be the indicator function of the setA. For random variableX, denoteX(c)=XI(|X| ≤c) for somec> . Denotelog+x=. ln max(e,x).

Candcdenote positive constants, which may be diﬀerent in various places.

Let{Xn,n≥}be a sequence of random variables deﬁned on a ﬁxed probability space (,F,P), and letSn=ni=Xifor eachn≥. Letnandmbe positive integers. WriteFm

n =

σ(Xi,nim). Givenσ-algebrasB,RinF, let

ψ(B,R) = sup

A∈B,B∈R,P(A)P(B)>

|P(AB) –P(A)P(B)|

P(A)P(B) . (.)

Deﬁne the mixing coeﬃcients by

ψ(n) =sup

k≥

ψFk,Fk+n, n≥.

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

ψ-mixing random variables ifψ(n)↓ asn→ ∞.

The concept ofψ-mixing random variables was introduced by Blumet al.[] and some applications have been found. See, for example, Blumet al.[] for strong law of large num-bers, Yang [] for almost sure convergence of weighted sums, Wu [] for strong consistency ofMestimator in linear model, Wanget al.[] for maximal inequality and Hájek-Rényi-type inequality, strong growth rate and the integrability of the supremum, Zhuet al.[] for

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strong convergence properties, Panet al.[] for strong convergence of weighted sums, and so on. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired. The main purpose of this pa-per is to establish the Khintchine-Kolmogorov-type convergence theorem, which can be applied to obtain the three series theorem and the Chung-type strong law of large num-bers forψ-mixing random variables. In addition, we will study the strong stability for weighted sums ofψ-mixing random variables, which generalizes the corresponding one of independent random variables.

For independent and identically distributed random variable sequences, Jamisonet al.

[] proved the following theorem.

Theorem A Let{X,Xn,n≥}be an independent and identically distributed sequence with the same distribution function F(x),and let{wn,n≥}be a sequence of positive numbers.

Write Wn=ni=ωnand N(x) =Card{n:Wn/ωnx},x> .If (i) Wn→ ∞andωnW–

n →asn→ ∞,

(ii) E|X|<∞andEN(|X|) <∞, (iii) x(y≥|x|N(y)/ydy)dF(x) <∞, then

Wn– n

i=

ωiXic a.s., (.)

where c is a constant.

The result of Theorem A for independent and identically distributed sequences has been generalized to some dependent sequences, such as negatively associated sequences, nega-tively superadditive dependent sequences,ρ˜-mixing sequences,ϕ˜-mixing sequences, and so forth. We will further study the strong stability for weighted sums ofψ-mixing random variables, which generalizes corresponding one of independent sequences. The main re-sults of the paper depend on the following important lemma - Rosenthal-type maximal inequality forψ-mixing random variables.

Lemma .(cf.Wanget al.[]) Let{Xn,n≥}be a sequence ofψ-mixing random variables satisfyingn=ψ(n) <∞,q≥.Assume that EXn= and E|Xn|q<∞for each n≥.Then there exists a constant C depending only on q andψ(·)such that

E

max

≤j≤n

a+j

i=a+

Xi

q

C

a+n

i=a+

E|Xi|q+ a+n

i=a+

EXiq/

(.)

for every a≥and n≥.In particular,we have

E

max

≤j≤n

j

i=

Xi

q

C

n

i=

E|Xi|q+

n

i=

EXiq/

(.)

for every n≥.

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Deﬁnition . A sequence{Xn,n≥} of random variables is said to be stochastically dominated by a random variableXif there exists a constantCsuch that

P|Xn|>x

CP|X|>x (.)

for allx≥ andn≥.

By the deﬁnition of stochastic domination and integration by parts, we can get the fol-lowing basic property for stochastic domination. For the proof, one can refer to Wanget al.[], Tang [] or Shen and Wu [].

Lemma . Let{Xn,n≥}be a sequence of random variables,which is stochastically dom-inated by a random variable X.For anyα> and b> ,the following statement holds

E|Xn|αI|Xn| ≤bCE|X|αI|X| ≤b+bαP|X|>b,

where C is a positive constant.

2 Khintchine-Kolmogorov-type convergence theorem

In this section, we will prove the Khintchine-Kolmogorov-type convergence theorem for ψ-mixing random variables. By using the Khintchine-Kolmogorov-type convergence the-orem, we can get the three series theorem and the Chung-type strong law of large numbers forψ-mixing random variables.

Theorem .(Khintchine-Kolmogorov-type convergence theorem) Let{Xn,n≥}be a sequence ofψ-mixing random variables satisfyingn=ψ(n) <∞.Assume that

n=

Var(Xn) <∞, (.)

thenn=(XnEXn)converges a.s.

Proof Without loss of generality, we assume thatEXn=  for alln≥. For anyε> , it can

be checked that

P

sup

k,m≥n

|SkSm|>ε

P

sup

k≥n|SkSn|>

ε

+P

sup

m≥n|SmSn|>

ε

≤ lim

N→∞P

max

n≤k≤N|SkSn|>

ε

≤ lim

N→∞

 (ε)

N

i=n+

Var(Xi)

=  ε

i=n+

Var(Xi)→, n→ ∞,

where the last inequality follows from Lemma .. Thus, the sequence{Sn,n≥}is a.s. Cauchy, and, therefore, we can obtain the desired result immediately. This completes the

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With the Khintchine-Kolmogorov-type convergence theorem in hand, we can get the three series theorem and the Chun-type strong law of large numbers forψ-mixing random variables.

Theorem .(Three series theorem) Let{Xn,n≥}be a sequence ofψ-mixing random variables satisfyingn=ψ(n) <∞.For some c> ,if

n=

P|Xn|>c<∞, (.)

n=

EXn(c) converges, (.)

n=

VarXn(c)<∞, (.)

thenn=Xnconverges almost surely.

Proof According to (.) and Theorem ., we have

n=

Xn(c)–EX(nc) converges a.s. (.)

It follows by (.) and (.) that

n=

X(nc) converges a.s. (.)

Obviously, (.) implies that

n=

PXn=Xn(c)=

n=

P|Xn|>c<∞. (.)

It follows by (.) and Borel-Cantelli lemma that

PXn=Xn(c), i.o.= . (.)

Finally, combining (.) with (.), we can get that∞n=Xnconverges a.s. The proof is

completed.

Theorem .(Chung-type strong law of large numbers) Let{Xn,n≥}be a sequence of

mean zeroψ-mixing random variables satisfyingn=ψ(n) <∞,and let{an,n≥}be a sequence of positive numbers satisfying <an↑ ∞.If there exists some p∈[, ]such that

n=

E|Xn|p apn

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then

lim

n→∞

an n

i=

Xi=  a.s. (.)

Proof It follows by (.) that

n=

Var(X(an)

n )

a

n

n=

E(X(an)

n ) a

n

=

n=

EX

nI(|Xn| ≤an) a

n

n=

E|Xn|p apn

<∞.

Therefore, we have by Theorem . that

n=

X(an)

nEXn(an)

an converges a.s. (.)

Sincep∈[, ], it follows byEXn=  that

n=

|EX(an)

n |

an

=

n=

|EXnI(|Xn| ≤an)| an

=

n=

|EXnI(|Xn|>an)|

an

n=

E|Xn|p apn

<∞,

which implies that

n=

EX(an)

n

an converges. (.)

Together with (.) and (.), we can see that

n=

X(an)

n

an converges a.s. (.)

By Markov’s inequality and (.), we have

n=

PXn=Xn(an)

=

n=

P|Xn|>an

n=

E|Xn|p apn

<∞. (.)

Hence, the desired result (.) follows from (.), (.), Borel-Cantelli lemma and

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3 Strong stability for weighted sums of

### ψ

-mixing random variables

In the previous section, we were able to get the Khintchine-Kolmogorov-type convergence theorem forψ-mixing random variables. In this section, we will study the strong stability for weighted sums ofψ-mixing random variables by using the Khintchine-Kolmogorov-type convergence theorem.

The concept of strong stability is as follows.

Deﬁnition . A sequence{Yn,n≥}is said to be strongly stable if there exist two con-stant sequences{bn,n≥}and{dn,n≥}with  <bn↑ ∞such that

b–nYndn→ a.s.

For the deﬁnition of strong stability, one can refer to Chow and Teicher []. Many au-thors have extended the strong law of large numbers for sequences of random variables to the case of triangular array of rowwise random variables and arrays of rowwise random variables. See, for example, Hu and Taylor [], Bai and Cheng [], Gan and Chen [], Kuczmaszewska [], Wu [–], Sung [], Wanget al.[–], Zhou [], Shen [], Shenet al.[], and so on.

Our main results are as follows.

Theorem . Let{an,n≥}and{bn,n≥}be two sequences of positive numbers with cn=bn/anand bn↑ ∞.Let{Xn,n≥}be a sequence ofψ-mixing random variables,which is stochastically dominated by a random variable X.Assume thatn=ψ(n) <∞.Denote N(x) =Card{n:cnx},x> , ≤p≤.If the following conditions are satisﬁed

(i) EN(|X|) <∞, (ii) tp–P(|X|>t)(

t N(y)/yp+dy)dt<∞, then there exist dnR,n= , , . . . ,such that

b–n n

i=

aiXidn→ a.s. (.)

Proof LetSn=ni=aiXi,Tn=ni=aiX(ci)

i . By Deﬁnition . and (i), we can see that

i=

PXi=Xi(ci)

=

i=

P|Xi|>ci

C

i=

P|X|>ci

CEN|X|<∞. (.)

By Borel-Cantelli lemma, for any sequence{dn,n≥} ⊂R, the sequences{b–

nTndn}

and {b–

nSndn} converge on the same set and to the same limit. We will show that b–

n

n

i=ai(X(ici)–EX

(ci)

i )→ a.s., which gives the theorem withdn=b–n

n

i=aiEX(ici).

Note that{ai(X(ci)

iEX

(ci)

i ),i≥}is a sequence of mean zeroψ-mixing random variables.

It follows fromCrinequality, Jensen’s inequality and Lemma . that

n=

E|an(X(cn)

nEXn(cn))|p bpn

C

n=

cnpE|Xn|pI

|Xn| ≤cn

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C

n=

cnpcpnP|X|>cn+E|X|pI|X| ≤cn

C

n=

P|X|>cn+C

n=

cnp

cn

tp–P|X|>tdt (.)

and

n=

cnp

cn

tp–P|X|>tdt

tp–P|X|>t n:cn≥t

cnpdt

C

tp–P|X|>t

t

N(y)/yp+dy

dt. (.)

The last inequality above follows from the fact that

n:cn≥t

cnp= lim

u→∞

n:t≤cn≤u

cnp

= lim

u→∞

u

t

ypdN(y)

= lim

u→∞

upN(u) –tpN(t) +p

u

t

y–(p+)N(y)dy

and

upN(u)≤p

u

y–(p+)N(y)dy→ asu→ ∞.

Obviously,

n=

P|X|>cn

EN|X|<∞. (.)

Thus, by (.)-(.) and condition (ii), we can see that

n=

E|an(X(cn)

nEXn(cn))|p bpn

<∞. (.)

Therefore,

b–n n

i=

aiX(ci)

iEX

(ci)

i

→ a.s.,

following from (.), Theorem . and Kronecker’s lemma immediately. The desired result

is obtained.

Corollary . Let the conditions of Theorem.be satisﬁed,and let EXn= for n≥. Assume thatEN(|X|/s)ds<∞.Then b–

n

n

i=aiXi→a.s.

Proof By Theorem ., we only need to prove that

b–n n

i=

aiEX(ci)

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In fact,

i=

ai|EXi(ci)| bi =

i=

c–i EXiI|Xi| ≤ci≤ ∞

i=

c–i E|Xi|I|Xi|>ci

i=

c–i

ciP

|Xi|>ci

+

ci

P|Xi|>t

dt

CEN|X|+C

EN|X|/sds<∞,

which implies (.) by Kronecker’s lemma. We complete the proof of the corollary.

Theorem . Let{an,n≥}and{bn,n≥}be two sequences of positive numbers with cn= bn/anand bn↑ ∞.Let{Xn,n≥}be a sequence of mean zeroψ-mixing random variables, which is stochastically dominated by a random variable X.Assume thatn=ψ(n) <∞.

Denote N(x) =Card{n:cnx},x> , ≤p≤.If the following conditions are satisﬁed (i) EN(|X|) <∞,

(ii) EN(|X|/s)ds<∞, (iii) max≤j≤ncpj

i=nc

p i =O(n), then

b–n n

i=

aiXi→ a.s. (.)

Proof By condition (i) and (.), we only need to prove thatb–

n

n

i=aiXi(ci)→ a.s. For

this purpose, it suﬃces to show that

b–n n

i=

ai

X(ci)

iEX

(ci)

i

→ a.s. (.)

and

b–n n

i=

aiEX(ci)

i → asn→ ∞. (.)

Equation (.) follows from the proof of Corollary . immediately.

To prove (.), we setε=  andεn=max≤j≤ncjforn≥. It follows fromCrinequality,

Jensen’s inequality and Lemma . that

n=

E|an(X(cn)

nEXn(cn))|p bpn

C

n=

cnpE|Xn|pI|Xn| ≤cn

C

n=

P|X|>cn

+C

n=

cnpE|X|pI|X| ≤cn

.

Obviously,

n=

P|X|>cn

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and

n=

cnpE|X|pI|X| ≤cn

n=

cnpE|X|pI|X| ≤εn

j=

εjpPεj–<|X| ≤εj

n=j

cnpC

j=

P|X|>εj–

C

 +

n=

P|X|>cn

C +EN|X|<∞.

Therefore,

n=

E|an(X(cn)

nEXn(cn))|p bpn

<∞, (.)

following from the statements above. By Theorem . and Kronecker’s lemma, we can

obtain (.) immediately. The proof is completed.

Theorem . Let{an,n≥}and{bn,n≥}be two sequences of positive numbers with cn=bn/anand bn↑ ∞.Let{Xn,n≥}be a sequence ofψ-mixing random variables,which is stochastically dominated by a random variable X.Assume thatn=ψ(n) <∞.Deﬁne N(x) =Card{n:cnx},R(x) =xN(y)y–dy,x> .If the following conditions are satisﬁed

(i) N(x) <∞for anyx> , (ii) R() =N(y)y–dy<,

(iii) EXR(|X|) <∞,

then there exist dnR,n= , , . . . ,such that

b–n n

i=

aiXidn→ a.s. (.)

Proof SinceN(x) is nondecreasing, then for anyx> 

R(x)≥N(x)

x

y–dy=  x

–N(x), (.)

which implies thatEN(|X|)≤EXR(|X|) <∞. Therefore,

i=

PXi=X(ci)

i

=

i=

P|Xi|>ci

C

i=

P|X|>ciCEN|X|<∞. (.)

By Borel-Cantelli lemma for any sequence{dn,n≥} ⊂R,{b–nSndn}and{b–nTndn}

converge on the same set and to the same limit. We will show that b–

n

n

i=ai(X (ci)

i

EX(ci)

i )→ a.s., which gives the theorem with dn=b–n

n

i=aiEX (ci)

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Lemma . that

n=

Var(anX(cn)

n )

b

n

n=

c–n EX(cn)

n

=

n=

c–n EXnI|Xn| ≤cn

CEN|X|+C

n=

c–n EXI|X| ≤cn

(.)

and

n=

c–n EXI|X| ≤cn=

n:cn≤

c–n EXI|X| ≤cn+

n:cn>

c–n EXI|X| ≤cn

.

=I+I. (.)

SinceN() =Card{n:cn≤} ≤R() <∞from (.) and (ii), it follows thatI<∞. For

I, we have

I=

n:cn>

c–n EXI|X| ≤cn

=

k=

k–<cn≤k

c–n EXI|X| ≤cn

k=

N(k) –N(k– )(k– )–EXI|X| ≤

+

k=

N(k) –N(k– )(k– )–EXI <|X| ≤k

.

=I+I,

I≤C

k=

N(k) –N(k– )

j=k–

j–=C

j=

j– j+

k=

N(k) –N(k– )

C

j=

(j+ )–N(j+ )≤C

y–N(y)dy<∞.

SinceN(x) is nondecreasing andR(x) is nonincreasing, we have

I≤

m=

EXIm–  <|X| ≤m

k=m

N(k)(k– )––k–

C

m=

EXIm–  <|X| ≤m

k=m

k+

k

N(x)x–dx

C

m=

EXR|X|Im–  <|X| ≤mCEXR|X|<∞.

Therefore,

n=

Var(anX(cn)

n )

b

n

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following from the above statements. By Theorem . and Kronecker’s lemma, we have

b–n n

i=

aiX(ci)

iEX

(ci)

i

→ a.s. (.)

Takingdn=b–n

n

i=aiEXi(ci), we haveb–n

n

i=aiX(ici)–dn→ a.s. The proof is

com-pleted.

Corollary . Let the conditions of Theorem . be satisﬁed. If EXn= , n≥ and

EN(|X|/s)ds<∞,then b–n

n

i=aiXi→a.s.

In the following, we denoteα(x) :R+→R+ as a positive and nonincreasing function

withan=α(n),bn=ni=ai,cn=bn/an,n≥, where

 <bn↑ ∞, (.)

 <lim inf

n→∞ n

–c

(logcn)≤lim sup

n→∞ n

–c

(logcn) <∞, (.)

log+x is nonincreasing forx> . (.)

Theorem . Let{Xn,n≥}be a sequence of identically distributedψ-mixing random variables withn=ψ(n) <∞.If E|X|α(log+|X|) <∞,then there exist dnR,n= , , . . . ,

such that b–

n

n

i=aiXidn→a.s.

Proof Sinceα(x) is positive and nonincreasing forx>  and  <bn↑ ∞, it follows that cn↑ ∞. By (.), we can choose constantsmN,C> ,C>  such that fornm,

Cncnα(logcn)≤Cn. (.)

Therefore, fornm, we have c

n

α(logcm)

Cn , which implies that

j=m c–j

j=m

α(logcm)

Cj

α(logcm)

Cm

. (.)

By (.)-(.), it follows that

j=m

E(ajXj(cj))

b

j

j=m c–j

{|X|≤cm–}

XdP+

j

i=m

{ci–<|X|≤ci}

XdP

C+

j=m c–j

j

i=m

{ci–<|X|≤ci}

XdP

C+C

i=m

i–α(logci)

{ci–<|X|≤ci}

XdP

C+C

i=m

α(logci)

{ci–<|X|≤ci}

|X|dP

C+C

i=m

{ci–<|X|≤ci}

|X|α

log+|X|

(12)

Therefore,

j=

Var(ajXj(cj))

b

j

j=

E(ajXj(cj))

b

j

<∞, (.)

which implies that

b–n n

i=

aiX(ci)

iEX

(ci)

i

→ a.s. (.)

from Theorem . and Kronecker’s lemma. By (.) and (.) again, we have

j=m

P|Xj|>cj

j=m P|Xj|α

log+|Xj|

cjα(logcj)

j=m P|X|α

log+|X|

Cj

<∞,

j=

PXj=Xj(cj)=

j=

P|Xj|>cj=

m–

j=

P|Xj|>cj+

j=m

P|Xj|>cj<∞.

By Borel-Cantelli lemma, we haveP(Xj=X(jcj), i.o.) = . Together with (.), we can see that

b–n n

i=

aiXiEX(ci)

i

→ a.s. (.)

Takingdn=b–n ni=aiEX(ci)

i forn≥, we get the desired result.

Theorem . Let {Xn,n ≥ } be a sequence of ψ-mixing random variables with

n=ψ(n) <∞.If for some≤p≤,

n=

npEXnαlog+|Xn|p<∞,

then there exist dnR,n= , , . . . ,such that b–n

n

i=aiXidn→a.s. Proof Similar to the proof of Theorem ., it is easily seen that

j=

PXj=Xj(cj)≤m–  +

j=m

P|Xj|αlog+|Xj|≥cjα(logcj)

m–  +

j=m

P|Xj|αlog+|Xj|≥Cj

<∞.

By Borel-Cantelli lemma for any sequence{dn,n≥} ⊂R, the sequences{b–n ni=aiXi

dn}and{b–

n

n

i=aiX (ci)

(13)

thatb–

n

n

i=ai(X (ci)

iEX

(ci)

i )→ a.s., which gives the theorem withdn=b–n

n

i=aiEX (ci)

i .

Note that{ai(X(ci)

iEX

(ci)

i )/bi,i≥}is a sequence of mean zeroψ-mixing random

vari-ables. ByCrinequality and Jensen’s inequality, we can see that

j=

E|aj(Xj(cj)–EX(jcj))|p

bpjC(m– ) +C

j=m

cjpE|Xj|pI|Xj| ≤cj

C(m– ) +C

j=m

j(logcj)pE|Xj|pI|Xj| ≤cj

C(m– ) +C

j=

jpEXjαlog+|Xj|p<∞.

It follows by Theorem . thatb–

n

n

i=ai(Xi(ci)–EX

(ci)

i )→ a.s. The proof is completed.

Corollary . Let the conditions of Theorem.be satisﬁed.Furthermore,suppose that EXn= andn=P(|Xn|>scn)ds<∞,then b–

n

n

i=aiXi→a.s.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1School of Mathematical Science, Anhui University, Hefei, 230601, P.R. China.2Department of Mathematics and Physics,

Anhui Jianzhu University, Hefei, 230022, P.R. China.

Acknowledgements

The authors are most grateful to the editor Andrei Volodin and an anonymous referee for careful reading of the manuscript and valuable suggestions, which helped in improving an earlier version of this paper. This work was supported by the Natural Science Project of Department of Education of Anhui Province (KJ2011z056) and the National Natural Science Foundation of China (11201001).

Received: 23 April 2013 Accepted: 24 July 2013 Published: 2 August 2013 References

1. Blum, JR, Hanson, DL, Koopmans, L: On the strong law of large numbers for a class of stochastic process. Z. Wahrscheinlichkeitstheor. Verw. Geb.2, 1-11 (1963)

2. Yang, SC: Almost sure convergence of weighted sums of mixing sequences. J. Syst. Sci. Math. Sci.15(3), 254-265 (1995)

3. Wu, QY: Strong consistency ofMestimator in linear model forρ-mixing,ϕ-mixing,ψ-mixing samples. Math. Appl.

17(3), 393-397 (2004)

4. Wang, XJ, Hu, SH, Shen, Y, Yang, WZ: Maximal inequality forψ-mixing sequences and its applications. Appl. Math. Lett.23, 1156-1161 (2010)

5. Zhu, YC, Deng, X, Pan, J, Ling, JM, Wang, XJ: Strong law of large numbers for sequences ofψ-mixing random variables. J. Math. Study45(4), 404-410 (2012)

6. Pan, J, Zhu, YC, Zou, WY, Wang, XJ: Some convergence results for sequences ofψ-mixing random variables. Chin. Q. J. Math.28(1), 111-117 (2013)

7. Jamison, B, Orey, S, Pruitt, W: Convergence of weighted averages of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.4, 40-44 (1965)

8. Wang, XJ, Hu, SH, Yang, WZ, Wang, XH: On complete convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables. Abstr. Appl. Anal.2012, Article ID 315138 (2012) 9. Tang, XF: Some strong laws of large numbers for weighted sums of asymptotically almost negatively associated

random variables. J. Inequal. Appl.2013, 4 (2013)

10. Shen, AT, Wu, RC: Strong and weak convergence for asymptotically almost negatively associated random variables. Discrete Dyn. Nat. Soc.2013, Article ID 235012 (2013)

(14)

12. Hu, TC, Taylor, RL: On the strong law for arrays and for the bootstrap mean and variance. Int. J. Math. Math. Sci.20(2), 375-382 (1997)

13. Bai, ZD, Cheng, PE: Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett.46, 105-112 (2000)

14. Gan, SX, Chen, PY: On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables. Acta Math. Sci., Ser. B27(2), 283-290 (2007)

15. Kuczmaszewska, A: On Chung-Teicher type strong law of large numbers forρ∗-mixing random variables. Discrete Dyn. Nat. Soc.2008, Article ID 140548 (2008)

16. Wu, QY: Complete convergence for negatively dependent sequences of random variables. J. Inequal. Appl.2010, Article ID 507293 (2010)

17. Wu, QY: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl.2010, Article ID 383805 (2010)

18. Wu, QY: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl.2012, 50 (2012)

19. Sung, SH: On the strong convergence for weighted sums of random variables. Stat. Pap.52, 447-454 (2011) 20. Wang, XJ, Li, XQ, Hu, SH, Yang, WZ: Strong limit theorems for weighted sums of negatively associated random

variables. Stoch. Anal. Appl.29, 1-14 (2011)

21. Wang, XJ, Hu, SH, Yang, WZ: Complete convergence for arrays of rowwise asymptotically almost negatively associated random variables. Discrete Dyn. Nat. Soc.2011, Article ID 717126 (2011)

22. Wang, XJ, Hu, SH, Volodin, AI: Strong limit theorems for weighted sums of NOD sequence and exponential inequalities. Bull. Korean Math. Soc.48(5), 923-938 (2011)

23. 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)

24. Wang, XJ, Hu, SH, Yang, WZ: Complete convergence for arrays of rowwise negatively orthant dependent random variables. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.106, 235-245 (2012)

25. 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)

26. Shen, AT: Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables. J. Inequal. Appl.2011, 93 (2011)

27. Shen, AT, Wu, RC, Chen, Y, Zhou, Y: Complete convergence of the maximum partial sums for arrays of rowwise of AANA random variables. Discrete Dyn. Nat. Soc.2013, Article ID 741901 (2013)

doi:10.1186/1029-242X-2013-360

References

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