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R E S E A R C H

Open Access

Complete convergence for weighted sums of

arrays of rowwise

ρ

˜

-mixing random variables

Aiting Shen, Ranchao Wu, Xinghui Wang and Yan Shen

*

*Correspondence:

shenyan@ahu.edu.cn School of Mathematical Science, Anhui University, Hefei, 230601, P.R. China

Abstract

Let{Xni,i≥1,n≥1}be an array of rowwise

ρ

˜-mixing random variables. Some sufficient conditions for complete convergence for weighted sums of arrays of rowwise

ρ

˜-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

ρ

˜-mixing random variables are obtained.

MSC: 60F15

Keywords:

ρ

˜-mixing sequence; arrays of rowwise

ρ

˜-mixing random variables; complete convergence; complete consistency

1 Introduction

The concept of complete convergence was introduced by Hsu and Robbins [] as follows. A sequence of random variables{Un,n≥}is said to converge completely to a constantC

if∞n=P(|UnC|>ε) <∞for allε> . In view of the Borel-Cantelli lemma, this implies

thatUnCalmost surely (a.s.). The converse is true if the{Un,n≥}are independent.

Hsu and Robbins [] 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 [] 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 [], Baum and Katz [], Gut [], Zarei [], and so forth. The main purpose of the paper is to provide complete convergence for weighted sums of arrays of rowwiseρ˜-mixing random variables. Firstly, let us recall the definitions of sequences ofρ˜-mixing random variables and arrays of rowwiseρ˜-mixing random variables.

Let{Xn,n≥}be a sequence of random variables defined on a fixed probability space

(,F,P). WriteFS=σ(Xi,iS⊂N). Given twoσ-algebrasB,RinF, let

ρ(B,R) = sup

XL(B),YL(R)

|EXYEXEY|

(VarXVarY)/.

Define theρ˜-mixing coefficients by

˜

ρ(k) =supρ(FS,FT) : finite subsetsS,T⊂Nsuch thatdist(S,T)≥k, k≥. Obviously, ≤ ˜ρ(k+ )≤ ˜ρ(k)≤, andρ˜() = .

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Definition . A sequence{Xn,n≥}of random variables is said to be aρ˜-mixing

se-quence if there existsk∈Nsuch thatρ˜(k) < .

An array{Xni,i≥,n≥}of random variables is called rowwiseρ˜-mixing random

vari-ables if for everyn≥,{Xni,i≥}is a sequence ofρ˜-mixing random variables. ˜

ρ-mixing random variables were introduced by Bradley [] and many applications have been found.ρ˜-mixing is similar toρ-mixing, but both are quite different. Many authors have studied this concept and provided interesting results and applications. See, for ex-ample, Bryc and Smolenski [], Peligrad [, ], Peligrad and Gut [], Utev and Peligrad [], Gan [], Cai [], Zhu [], Wu and Jiang [, ], An and Yuan [], Kuczmaszewska [], Sung [], Wanget al.[–], and so on.

Recently, An and Yuan [] obtained a complete convergence result for weighted sums of identically distributedρ˜-mixing random variables as follows.

Theorem . Let p> /αand/ <α≤.Let{Xn,n≥}be a sequence of identically

dis-tributedρ˜-mixing random variables with EX= .Assume that{ani, ≤in,n≥}is an

array of real numbers satisfying

n

i=

|ani|p=O

for some <δ< , (.)

Ank=

≤in:|ani|p> (k+ )–

ne–/k, ∀k≥,n≥. (.)

Then the following statements are equivalent:

(i) E|X|p<∞; (ii) ∞n=npα–P(max

≤jn|

j

i=aniXi|>εnα) <∞for allε> .

Sung [] pointed out that the array{ani, ≤in,n≥}satisfying both (.) and (.)

does not exist and obtained a new complete convergence result for weighted sums of iden-tically distributedρ˜-mixing random variables as follows.

Theorem . Let p> /α and/ <α≤.Let {Xn,n≥} be a sequence of identically

distributedρ˜-mixing random variables with EX= .Assume that{ani, ≤in,n≥}is

an array of real numbers satisfying

n

i=

|ani|q=O(n) for some q>p. (.)

If E|X|p<∞,then

n=

npα–P

max

≤jn

j

i=

aniXi

>εnα

<∞, ∀ε> . (.)

Conversely,if(.)holds for any array{ani, ≤in,n≥}satisfying(.),then E|X|p<∞.

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of rowwiseρ˜-mixing random variables under mild conditions. The main idea is inspired by Baeket al.[] and Wu []. As applications, the results of Baum and Katz [] from the i.i.d. case to the arrays of rowwiseρ˜-mixing setting are obtained. The Marcinkiewicz-Zygmund type strong law of large numbers for sequences ofρ˜-mixing random variables is provided. We give some sufficient conditions for complete convergence for weighted sums of arrays of rowwiseρ˜-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 symbolC denotes a positive constant which is not nec-essarily the same one in each appearance and xdenotes 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. Denotelogx=ln max(x,e),X+=max(X, ) and

X=max(–X, ).

The paper is organized as follows. Two important lemmas are provided in Section . The main results and their proofs are presented in Section . We get complete convergence for arrays of rowwiseρ˜-mixing random variables which are stochastically dominated by a random variableX.

2 Preliminaries

Firstly, we give the definition of stochastic domination.

Definition . A sequence{Xn,n≥} of random variables 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{Xni,i≥,n≥}of rowwise random variables 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≥.

The proofs of the main results of the paper are based on the following two lemmas. One is the classic Rosenthal type inequality forρ˜-mixing random variables obtained by Utev and Peligrad [], the other is the fundamental inequalities for stochastic domination.

Lemma .(cf.Utev and Peligrad [, Theorem .]) Let{Xn,n≥}be a sequence ofρ˜

-mixing random variables,EXi= ,E|Xi|p<∞for some p≥and for every i≥.Then

there exists a positive constant C depending only on p such that

E

max

≤jn

j

i=

Xi

pC

n

i=

E|Xi|p+

n

i=

EXi

p/

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

stochastically dominated by a random variable X.For anyα> and b> ,the following two statements hold:

E|Xni|αI

|Xni| ≤b

C

E|X|αI|X| ≤b+bαP|X|>b, (.)

E|Xni|αI

|Xni|>b

CE|X|αI

|X|>b, (.)

where Cand Care positive constants.

Proof The proof of this lemma can be found in Wu [] or Wanget al.[].

3 Main results and their applications

In this section, we provide complete convergence for weighted sums of arrays of row-wiseρ˜-mixing random variables. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of ρ˜-mixing random variables are obtained. Let{Xni,i≥,n≥} be an array of rowwiseρ˜-mixing

random variables. We assume that the mixing coefficients ρ˜(·) in each row are the same.

Theorem . Let{Xni,i≥,n≥} be an array of rowwiseρ˜-mixing random variables

which is stochastically dominated by a random variable X and EXni= for all i≥,n≥, β≥–.Let{ani,i≥,n≥}be an array of constants such that

sup

i≥

|ani|=O

nr for some r>  (.)

and

i=

|ani|=O

for someα∈[,r). (.)

Assume further that +α+β> and there exists someδ> such thatα/r+  <δ≤and s=max( ++αr+β,δ).If E|X|s<∞,then for allε> ,

n=

P

max

≤jn

j

i=

aniXni

>ε

<∞. (.)

If +α+β< and E|X|<∞,then(.)still holds for allε> .

Proof Without loss of generality, we assume thatani>  for alli≥ andn≥

(other-wise, we usea+

niandaniinstead ofanirespectively, and note thatani=a+niani). From the

conditions (.) and (.), we assume that

sup

i≥

ani=nr,

i=

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If  +α+β<  andE|X|<∞, then the result can be easily proved by the following:

n=

P

max

≤jn

j

i=

aniXni

>εCn=

E

max

≤jn

j

i=

aniXni

Cn= n i=

E|aniXni|

C

n=

+βE|X|<.

In the following, we consider the case of  +α+β> . Denote

Xni =aniXniI

|aniXni| ≤

, i≥,n≥.

It is easy to check that for anyε> ,

max

≤jn

j

i=

aniXni

>ε

⊂max

≤in|aniXni|> 

max

≤jn

j i= Xni >ε ,

which implies that

P

max

≤jn

j

i=

aniXni

>ε

Pmax

≤in|aniXni|> 

+P

max

≤jn

j i= Xni >εn i=

P|aniXni|> 

+P

max

≤jn

j

i=

XniEXni

>ε–max

≤jn

j i= EXni . (.)

Firstly, we show that

max

≤jn

j

i=

EXni

→ asn→ ∞. (.)

Actually, by the conditionsEXni= , Lemma ., (.) andE|X|+α/r<∞(sinceE|X|s<∞),

we have that

max

≤jn

j i= EXni = max

≤jn

j

i=

EaniXniI

|aniXni| ≤

= max

≤jn

j

i=

EaniXniI

|aniXni|> 

n

i=

E|aniXni|+α/rI

x|aniXni|> 

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C

n

i=

a+niα/rE|X|+α/rI

|X|> 

ani

Csup

i≥

ani

α/rn

i=

aniE|X|+α/rI

|X|>nr

Cnrα/rnαE|X|+α/rI|X|>nr

=CE|X|+α/rI|X|>nr→ asn→ ∞,

which implies (.). It follows from (.) and (.) that fornlarge enough,

P

max

≤jn

j

i=

aniXni

>ε

n

i=

P|aniXni|> 

+P

max

≤jn

j

i=

XniEXni

>ε

.

Hence, to prove (.), we only need to show that

I=.

n=

n

i=

P|aniXni|> 

<∞ (.)

and

J=.

n=

nβP

max

≤jn

j

i=

XniEXni

>ε

<∞. (.)

By (.) andE|X|s<∞, we can get that

n=

n

i=

P|aniXni|> 

C

n=

n

i=

P|aniX|> 

C

n=

n

i=

aniE|X|I

|X|> 

ani

C

n=

+βE|X|I|X|>nr

C

n=

+β

k=n

E|X|Ikr≤ |X|< (k+ )r

=C

k= k

n=

+βE|X|Ikr≤ |X|< (k+ )r

C

k=

k+α+βE|X|Ikr≤ |X|< (k+ )r

C

k=

E|X|+(+α+β)/rIkr≤ |X|< (k+ )r

CE|X|+(+α+β)/r<∞,

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By Markov’s inequality, Lemma .,Cr’s inequality and Jensen’s inequality, we have for

M≥ that

n=

nβP

max

≤jn

j

i=

XniEXni

>ε  ≤Cn=

nβE

max

≤jn

j

i=

XniEXni

MCn= n i=

E Xni

M/

+

n

i=

E Xni M

.

=J+J. (.)

Take

M>max

, ( +β)

r[δ– ( +α/r)],  +

 +α+β

r

,

which implies thatβr[δ– ( +α/r)]M/ < – andα+βr(M– ) < –. SinceE|X|δ<,

we have by Lemma ., Markov’s inequality and (.) that

J =. C

n= n i=

E Xni

M/ =Cn= n i=

E|aniXni|I

|aniXni| ≤

M/ ≤Cn= n i=

P|aniX|> 

+

n

i=

E|aniX|I

|aniX| ≤

M/ ≤Cn= n i=

niE|X|δ

M/

sinceδ≤

C

n=

sup

i≥

ani

δ–n

i= ani M/ ≤Cn=

nβnr(δ–)·nαM/

=C

n=

r[δ–(+α/r)]M/<∞. (.)

By Lemma . again, we can see that

J =. C

n= n i=

E Xni M

=Cn= n i=

E|aniXni|MI

|aniXni| ≤

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Cn= n i=

P|aniX|> 

+Cn= n i=

E|aniX|MI

|aniX| ≤

.

=J+J. (.)

J<∞has been proved by (.). In the following, we show thatJ<∞. Denote

Inj=

i: (nj)r≤/ani<

n(j+ )r, n≥,j≥. (.)

It is easily seen thatInkInj=∅fork=jand

j=Inj=Nfor alln≥. Hence,

J=C

n= n i=

E|aniX|MI

|aniX| ≤

Cn= j=

iInj

E|aniX|MI

|aniX| ≤

Cn= j=

(Inj)(nj)–rME|X|MI

|X| ≤n(j+ )r

Cn= j=

(Inj)(nj)–rM n(j+)

k=

E|X|MIk≤ |X|r <k+ 

=Cn= j=

(Inj)(nj)–rMn

k=

E|X|MIk≤ |X|r <k+ 

+Cn= j=

(Inj)(nj)–rM n(j+)

k=n+

E|X|MIk≤ |X|r <k+ 

.

=J+J. (.)

It is easily seen that for allm≥, we have that

=

i=

ani=

j=

iInj

ani

j=

(Inj)

n(j+ )–r

j=m

(Inj)

n(j+ )–r

j=m

(Inj)

n(j+ )–r

n(m+ )

n(j+ )

r(M–)

=

j=m

(Inj)

n(j+ )–rMn(m+ )r(M–),

which implies that for allm≥,

j=m

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Therefore,

J=. C

n=

j=

(Inj)(nj)–rMn

k=

E|X|MIk≤ |X|r <k+ 

C

n=

·r(M–)

n

k=

E|X|MIk≤ |X|r <k+ 

C

k=

n=

+βr(M–)E|X|MIk≤ |X|r <k+ 

+C

k=

n=k/

+βr(M–)E|X|MIk≤ |X|r <k+ 

C+C

k=

k+α+βr(M–)E|X|MIk≤ |X|r <k+ 

C+C

k=

E|X|M++αr+β–(M–)Ik≤ |X|r <k+ 

C+CE|X|++αr+β <sinceE|X|s<(.)

and

J=. C

n=

j=

(Inj)(nj)–rM n(j+)

k=n+

E|X|MIk≤ |X|r <k+ 

C

n=

k=n+

jkn–

(Inj)(nj)–rME|X|MI

k≤ |X|r <k+ 

C

n=

k=n+

r(M–)

k n

r(M–)

E|X|MIk≤ |X|r <k+ 

C

k= k/

n=

+β·kr(M–)E|X|MIk≤ |X|r <k+ 

C

k=

k+α+βr(M–)E|X|MIk≤ |X|r <k+ 

C

k=

E|X|M++αr+β–(M–)Ik≤ |X|r <k+ 

CE|X|++αr+β <sinceE|X|s<. (.)

Thus, the inequality (.) follows from (.)-(.), (.), (.) and (.). This completes the proof of the theorem.

Theorem . Let{Xni,i≥,n≥}be an array of rowwise ρ˜-mixing random variables

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Let{ani,i≥,n≥}be an array of constants such that(.)holds and

i=

|ani|=O(). (.)

If E|X|log|X|<∞,then for allε> ,

n=

n–P

max

≤jn

j

i=

aniXni

>ε

<∞. (.)

Proof We use the same notations as those in Theorem .. According to the proof of The-orem ., we only need to show that (.) and (.) hold, whereβ= – andα= .

The factE|X|log|X|<∞yields that

I =.

n=

n–

n

i=

P|aniXni|> 

C

n=

n–

n

i=

P|aniX|> 

C

k= k

n=

n–E|X|Ikr≤ |X|< (k+ )r

C

k=

logkE|X|Ikr≤ |X|< (k+ )r

C

k=

E|X|log|X|Ikr≤ |X|< (k+ )r

CE|X|log|X|<∞,

which implies (.) forβ= –.

By Markov’s inequality, Lemmas . and ., we can get that

J =.

n=

n–P

max

≤jn

j

i=

XniEXni

>ε

C

n=

n–

n

i=

E Xni

=C

n=

n–

n

i=

E|aniXni|I

|aniXni| ≤

C

n=

n–

n

i=

P|aniX|> 

+C

n=

n–

n

i=

E|aniX|I

|aniX| ≤

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Here,J*

andJ*areJandJwhenM= , respectively. Similar to the proof ofJ, we can get

that

J*≤C+CE|X|<∞. (.)

Similar to the proof ofJ, we have

J*≤C

k= [k/]

n=

n–·krE|X|Ik≤ |X|r <k+ 

C

k=

logk·kr·krE|X|Ik≤ |X|r <k+ 

CE|X|log|X|<∞. (.)

This completes the proof of the theorem from the statements above. By Theorems . and ., we can extend the results of Baum and Katz [] for independent and identically distributed random variables to the case of arrays of rowwise ρ˜-mixing random variables as follows.

Corollary . Let{Xni,i≥,n≥}be an array of rowwiseρ˜-mixing random variables

which is stochastically dominated by a random variable X and EXni= for all i≥,n≥. (i) Letp> and≤t< .IfE|X|pt<,then for allε> ,

n=

np–P

max

≤jn

j

i=

Xni

>εnt

<∞. (.)

(ii) IfE|X|log|X|<∞,then for allε> ,

n=

nP

max

≤jn

j

i=

Xni

>εn

<∞. (.)

Proof (i) Letani=  ifi>nandani=n–/tifin. Hence, conditions (.) and (.) hold

forr= /tandα=  – /t<r.β=. p–  > –. It is easy to check that

 +α+β=p–

t> ,  +

 +α+β

r =pt

.

=s, α

r +  =t<pt

. =s.

Therefore, the desired result (.) follows from Theorem . immediately.

(ii) Letani=  ifi>nandani=n–ifin. Hence, conditions (.) and (.) hold for

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Theorem . Let{Xn,n≥}be a sequence ofρ˜-mixing random variables which is

stochas-tically dominated by a random variable X and EXn= for n≥. (i) Letp> and≤t< .IfE|X|pt<∞,then for allε> ,

n=

np–P

max

≤jn

j

i=

Xi

>εnt

<∞. (.)

(ii) IfE|X|log|X|<∞,then for allε> ,

n=

nP

max

≤jn

j

i=

Xi

>εn

<∞. (.)

By Theorem ., we can get the Marcinkiewicz-Zygmund type strong law of large num-bers forρ˜-mixing random variables as follows.

Corollary . Let {Xn,n≥} be a sequence of ρ˜-mixing random variables which is

stochastically dominated by a random variable X and EXn= for n≥. (i) Letp> and≤t< .IfE|X|pt<∞,then

n–t n

i=

Xi→ a.s.,n→ ∞. (.)

(ii) IfE|X|log|X|<∞,then

n

n

i=

Xi→ a.s.,n→ ∞. (.)

Proof (i) By (.), we can get that for allε> ,

∞>

n=

np–P

max

≤jn

j

i=

Xi

>εnt

=

k= k+–

n=k

np–P

max

≤jn

j

i=

Xi

>εnt

k=(k)p–kP(max≤j≤k|ji=Xi|>εk+

t ) ifp≥,

k=(k+)p–kP(max≤j≤k|ji=Xi|>εk+

t ) if  <p< , ≥

k=P(max≤j≤k|

j

i=Xi|>εk+

t ) ifp,

k=P(max≤j≤k|

j

i=Xi|>εk+

t ) if  <p< .

By Borel-Cantelli lemma, we obtain that

maxjk|ji=Xi|

k+t

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For all positive integersn, there exists a positive integerksuch that k–≤n< k. We

have by (.) that

|n i=Xi|

nt

≤ max

k–n<k |n

i=Xi|

nt ≤

t max

≤j≤k|

j

i=Xi|

k+t

→ a.s.,k→ ∞,

which implies (.).

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

Remark . We point out that the cases  +α+β>  and  +α+β<  are considered in Theorem . and the case  +α+β =  is considered in Theorem ., respectively. Theorem . and Theorem . consider the complete convergence for weighted sums of arrays of rowwiseρ˜-mixing random variables, while Theorem . considers the complete convergence for weighted sums of sequences ofρ˜-mixing random variables. In addition, Theorem . and Theorem . could be applied to obtain the Baum and Katz type result for arrays of rowwiseρ˜-mixing random variables, while Theorem . could be applied to establish the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of

˜

ρ-mixing random variables.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors are most grateful to the editor Jewgeni Dshalalow and anonymous referees 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 National Natural Science Foundation of China (11201001, 11171001), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, 1208085QA03), the 211 project of Anhui University, the Youth Science Research Fund of Anhui University, Applied Teaching Model Curriculum of Anhui University (XJYYXKC04) and the Students Science Research Training Program of Anhui University (KYXL2012007, kyxl2013003).

Received: 15 March 2013 Accepted: 22 July 2013 Published: 29 July 2013 References

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References

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