Complete convergence and complete moment convergence for a class of random variables

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variables

Xinghui Wang and Shuhe Hu

*

*Correspondence:

hushuhe@263.net

School of Mathematical Science, Anhui University, Hefei, 230039, P.R. China

Abstract

In this paper, we establish the complete convergence and complete moment convergence and obtain the equivalence of the complete convergence and complete moment convergence for the class of random variables satisfying a Rosenthal-type maximal inequality. Baum-Katz-type theorem and Hsu-Robbins-type theorem are extended to the case of this class of random variables.

MSC: 60F15

Keywords: complete convergence; complete moment convergence;

Baum-Katz-type theorem; Hsu-Robbins-type theorem

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 constant Cif∞n=P(|UnC|>ε) <∞for allε> . In view of the Borel-Cantelli lemma, this im-plies thatUnCalmost surely (a.s.). The converse is true if the{Un,n≥}are indepen-dent. 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-Robbin-Erdös is a fundamental theorem in probability theory which has been gen-eralized and extended in several directions by many authors. One of the most important generalizations is that by Baum and Katz [] for the strong law of large numbers as follows.

Theorem A(Baum and Katz []) Letα> /andαp> .Let{Xn,n≥}be a sequence if i.i.d.random variables.Assume further that EX= ifα≤.Then the following statements

are equivalent:

(i) E|X|p<∞;

(ii) ∞

n=

nαp–P

max

≤j≤n

j

i=

Xi

>εnα

<∞, for allε> .

Many authors studied the Baum-Katz-type theorem for dependent random variables; see, for example, Peligrad [] for a strong stationaryρ-mixing sequence, Peligrad and Gut

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[] for aρ*-mixing sequence, Stoica [, ] for a martingale diﬀerence sequence, Stoica []

for bounded subsequences, Wang and Hu [] forϕ-mixing random variables, and so forth. One of the most interesting inequalities to probability theory is the Rosenthal-type max-imal inequality. For a sequence{Xi, ≤in}of i.i.d. random variables withE|X|q<∞

for someq≥, there exist positive constantsCqdepending only onqsuch that

E

max

≤j≤n

j

i=

(Xi–EXi)

q

Cq

n

i=

E|Xi|q+

n

i=

EXi

q/

.

The inequality above has been obtained for dependent random variables by many authors. See, for example, Shao [] for negatively associated random variables, Utev and Peligrad [] forρ*-mixing random variables, Wanget al.[] forϕ-mixing random variables with

the mixing coeﬃcients satisfying certain conditions, and so forth.

The purpose of this work is to obtain complete convergence and complete moment con-vergence for a sequence of random variables satisfying a Rosenthal-type maximal inequal-ity.

Throughout the paper, let{Xn,n≥}be a sequence of random variables deﬁned on a ﬁxed probability space (,A,P). LetI(A) be the indicator function of the setA. Denote Sn=ni=Xi,S= ,ln+x=ln max(x,e),x+=xI(x≥). The symbolCdenotes a positive

constant which may be diﬀerent in various places.

The following deﬁnition will be used frequently in the paper.

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|>xCP |X|>x

for allx≥ andn≥.

In this paper, the present investigation is to provide the complete convergence and complete moment convergence for a sequence of the class of random variables satis-fying a Rosenthal-type maximal inequality and prove the equivalence of the complete convergence and complete moment convergence. Baum-Katz-type theorem and Hsu-Robbins-type theorem are extended to the case of the class of random variables satisfying a Rosenthal-type maximal inequality. As a result, the Marcinkiewicz-Zygmund strong law of large numbers for the class of random variables is obtained.

2 Main results

Theorem . Letα> /,αp≥and p> .Suppose that{Xn,n≥}is a sequence of zero mean random variables which is stochastically dominated by a random variable X with E|X|p<.Assume that for any q,there exists a positive constant Cqdepending only on q such that

E

max

≤j≤n

j

i=

(Yti–EYti)

q

Cq

n

i=

E|Yti|q+

n

i=

EYti

q/

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where Yti= –tI(Xi< –t) +XiI(|Xi| ≤t) +tI(Xi>t)for all t> .Then

n=

nαp–P

max

≤j≤n|Sj| ≥εn

α

<∞, for allε>  (.)

and

n=

nαp––αE

max

≤j≤n|Sj|–εn

α+

<∞, for allε> . (.)

Furthermore, (.)is equivalent to(.).

Corollary . Let <p< .Suppose that{Xn,n≥}is a sequence of zero mean random variables which is stochastically dominated by a random variable X with E|X|p<. As-sume further that(.)holds,then

Sn

n/p → a.s.

Forp= , we have the following theorem.

Theorem . Letα> .Suppose that{Xn,n≥}is a sequence of zero mean random vari-ables which is stochastically dominated by a random variable X with E|X|ln+|X|<∞. Assume further that(.)holds,then for allε> ,

n=

n–E

max

≤j≤n|Sj|–εn

α+

<∞. (.)

Theorem . Letα> /,p> andαp> .Suppose that{Xn,n≥}is a sequence of zero mean random variables which is stochastically dominated by a random variable X with E|X|p<.Assume further that(.)holds,then for allε> ,

n=

nαp–P

sup j≥n

Sj

ε

<∞. (.)

Remark . In (.),{Yti,i≥}is a monotone transformation of{Xi,i≥}. If{Xi,i≥} is a sequence of independent random variables, then (.) is clearly satisﬁed. There are many sequences of dependent random variables satisfying (.) for allq≥. Examples in-clude sequences of NA random variables (see Shao []),ρ*-mixing random variables (see Utev and Peligrad []),ϕ-mixing random variables with the mixing coeﬃcients satisfy-ing certain conditions (see Wanget al.[]),ρ–-mixing random variables with the mixing coeﬃcients satisfying certain conditions (see Wang and Lu []).

Remark . In Theorem ., we not only generalize the Baum-Katz-type theorem for the

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3 Lemmas

In this section, the following lemmas are very useful to prove the main results of the paper.

Lemma . (cf. Wu []) Let {Xn,n≥} be a sequence of random variables, which is

stochastically dominated by a random variable X.Then for any a> and b> ,the follow-ing two statements hold:

E|Xn|aI |Xn| ≤bC

E|X|aI |X| ≤b+baP |X|>b and

E|Xn|aI |Xn|>b

CE|X|aI |X|>b

,

where Cand Care positive constants.

Lemma . Under the conditions of Theorem.,

n=

nαp––α

P

max

≤j≤n|Sj|>t

dt<∞. (.)

Proof For ﬁxedn≥, denoteYti=XiYti,i≥. Then it follows that

n=

nαp––α

Pmax

≤j≤n|Sj|>t

dt

n=

nαp––α

P

max

≤j≤n

j

i=

(Yti–EYti)

>t/

dt

+ ∞

n=

nαp––α

P

max

≤j≤n

j

i=

YtiEYti

>t/

dt

:=I+J.

ForJ, noting that|Yti| ≤ |Xi|I(|Xi|>t), we have by Markov’s inequality, Lemma . and E|X|p<∞that

JC

n=

nαp––α

t– n

i=

EYtidt

C

n=

nαp––α

t –

n

i=

E|Xi|I |Xi|>tdt

C

n=

nαp––α

t

–E|X|I |X|>tdt

=C

n=

nαp––α

m=n

(m+)α

t

–E|X|I |X|>tdt

C

n=

nαp––α

m=n

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=C

m=

m–E|X|I |X|> m

n=

nαp––α

C

m=

m–E|X|I |X|>mαmαpα

=C

m=

mαp––αE|X|I |X|>

=C

m=

mαp––α

n=m

E|X|I n<|X|/αn+ 

C

n=

E|X|I n<|X|/αn+ 

n

m=

mαp––α

C

n=

nαpαE|X|I n<|X|/αn+ ≤CE|X|p<∞. (.)

ForI, by Markov’s inequality and (.), we have that forq≥,

ICq

n=

nαp––α

t

qE

max

≤j≤n

j

i=

(Yti–EYti)

q dt

Cq

n=

nαp––α

tq n

i=

E|Yti|qdt+Cq

n=

nαp––α

tq

n

i=

EYti

q/

dt

:=I+I. (.)

We will consider the following three cases: Case .α> /,αp>  andp≥.

Takingq>max(p,ααp–

), which implies thatαp–  –αq+q/ < –. We have by Lemma .

and the proof of (.) that

I≤Cq

n=

nαp––α

tq n

i=

E|Xi|qI |Xi| ≤t

+tqP |Xi|>t

dt

C

n=

nαp––α

t

qE|X|qI |X| ≤tdt+C

n=

nαp––α

t

–E|X|I |X|>tdt

C

n=

nαp––α

m=n

(m+)α

tqE|X|qI |X| ≤tdt+C

C

n=

nαp––α

m=n

––αqE|X|qI |X| ≤(m+ )α+C

=C

m=

––αqE|X|qI |X| ≤(m+ )α m

n=

nαp––α+C

C

m=

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+C

m=

mαp––αqE|X|qI |X| ≤+C

C

m=

m–E|X|pI <|X| ≤(m+ )α

+C

m=

(pq)– m

j=

jαqP j–  <|X|/αj+C

CE|X|p+C

j=

jαqP j–  <|X|/αj

m=j

(pq)–+C

C

j=

jαpP j–  <|X|/αj+CCE|X|p+C<. (.)

Note thatEX<ifE|X|p<forp. We have that

I≤C

n=

nαp––α

t

q

n

i=

EXi

q/

dt

C

n=

nαp––α

t

q

n

i=

EX

q/

dt

C

n=

nαp––α+q/

tqdt

C

n=

nαp––αq+q/<∞.

Case .α> /,αp>  and  <p< .

Takeq= . Similar to the proofs of (.) and (.), we have that

IC

n=

nαp––α

t– n

i=

EXiI |Xi| ≤t

+tP |Xi|>t

dt

C

n=

nαp––α

m=n

mα–EXI |X| ≤(m+ )α+C

=C

m=

mα–EXI |X| ≤(m+ )α

m

n=

nαp––α+C

C

m=

mαp––αEXI <|X| ≤(m+ )α

+C

m=

mαp––αEXI |X| ≤+C

C

m=

m–E|X|pI <|X| ≤(m+ )α

+C

m=

(p–)– m

j=

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CE|X|p+C

j=

jαP j–  <|X|/αj

m=j

(p–)–+C

C

j=

jαpP j–  <|X|/αj+C

CE|X|p+C<∞. (.)

Case .α> /,αp=  andp> .

Takeq= . Note that / <α<  ifαp= . Similar to the proofs of (.), it follows that I<∞. From the statements above, (.) is proved. The proof of the lemma is completed.

Lemma .(cf.Sung []) Let{Yn,n≥}and{Zn,n≥}be sequences of random variables. Then for any q> ,ε> and a> ,

E

max

≤j≤n

j

i=

(Yi+Zi)

εa

+ ≤

εq +

q– 

aq–Emax≤j≤n

j

i=

Yi

q

+Emax

≤j≤n

j

i=

Zi

.

4 The proofs of main results

Proof of Theorem . First, we prove (.). For ﬁxedn≥, let Xni = –nαI(Xi< –nα) +

XiI(|Xi| ≤) +nαI(Xi>) andXni =XiXni,i≥. Then it is easy to have that

n=

nαp–P

max

≤j≤n|Sj| ≥εn

α

n=

nαp–P

max

≤j≤n

j

i=

(XniEXni)

εnα/

+ ∞

n=

nαp–P

max

≤j≤n

j

i=

XniEXni

εnα/

:=I*+J*.

ForJ*, noting that|X

ni| ≤ |Xi|I(|Xi|>), we have by Markov’s inequality, Lemma . and the proof of (.) that

J*≤C

n=

nαp––α

n

i=

EXni

C

n=

nαp––α n

i=

E|Xi|I |Xi|>

C

n=

nαp––αE|X|I |X|>

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ForI*, by Markov’s inequality and (.), we have that for anyq,

I*≤Cq

n=

nαp––αqE

max

≤j≤n

j

i=

(Xni–EXni)

q

Cq

n=

nαp––αq n

i=

E|Xni|q+Cq

n=

nαp––αq

n

i=

EXni

q/

:=I*+I*. (.)

We consider the following three cases: Case .α> /,αp>  andp≥. Takeq>max(p,ααp–

), which implies thatαp–  –αq+q/ < –.

ForI*

, we have byCr’s inequality, the proofs of (.) and (.) that

I*≤C

n=

nαp––αq n

i=

E|Xi|qI |Xi| ≤

+nαqP |X i|>

C

n=

nαp––αq n

i=

E|X|qI |X| ≤+nαqP |X|>

C

n=

nαp––αqE|X|qI |X| ≤+C

n=

nαp––αE|X|I |X|>

<∞. (.)

ForI*

, note thatEX<∞ifE|X|p<∞forp≥. We have that

I*≤C

n=

nαp––αq

n

i=

EXi

q/ ≤C

n=

nαp––αq

n

i=

EX

q/

C

n=

nαp––αq+q/<∞.

Case .α> /,αp>  and  <p< .

Takeq= . Similar to the proofs of (.), (.), (.) and (.), we have that

I*≤C

n=

nαp––α n

i=

EXiI |Xi| ≤+nαP |Xi|>

C

n=

nαp––αEXI |X| ≤+C

n=

nαp––αE|X|I |X|>

<∞. (.)

Case .α> /,αp=  andp> .

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Next, we prove (.). SinceSj=ji=XiandXi= (Xni–EXni) + (XniEXni),i= , , . . . ,n. By Lemma ., the proofs of (.) andI*<, it follows that

n=

nαp––αEmax

≤j≤n|Sj|–εn

α+

C

n=

nαp––αqE

max

≤j≤n

j

i=

(Xni–EXni)

q

+ ∞

n=

nαp––αE

max

≤j≤n

j

i=

XniEXni

C

n=

nαp––αqE

max

≤j≤n

j

i=

(Xni–EXni)

q

+C

n=

nαp––α n

i=

E|Xi|I |Xi|> <∞.

Hence (.) holds.

We will prove the equivalence of (.) and (.). First, we prove that (.) implies (.). In fact, for allε> , we have by Lemma . that

n=

nαp––αE

max

≤j≤n|Sj|–εn

α+

= ∞

n=

nαp––α

Pmax

≤j≤n|Sj|–εn

α+>tdt

= ∞

n=

nαp––α

P

max

≤j≤n|Sj|–εn

α

>t

dt

= ∞

n=

nαp––α

P

max

≤j≤n|Sj|–εn

α

>t

dt

+ ∞

n=

nαp––α

Pmax

≤j≤n|Sj|–εn

α>tdt

n=

nαp–P

max

≤j≤n|Sj| ≥εn

α

+ ∞

n=

nαp––α

P

max

≤j≤n|Sj|>t

dt<∞.

Next, we prove that (.) implies (.). It is easy to see that

n=

nαp––αE

max

≤j≤n|Sj|–εn

α+

= ∞

n=

nαp––α

P

max

≤j≤n|Sj|–εn

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n=

nαp––α

εnα

P

max

≤j≤n|Sj|>εn

α

+t

dt

ε

n=

nαp–P

max

≤j≤n|Sj| ≥εn

α

. (.)

Hence, by (.), (.) implies (.). The proof of the theorem is completed.

Proof of Corollary. Takingαp=  in Theorem ., we have that for allε> ,

n=

n–P

max

≤j≤n|Sj| ≥εn

/p<.

The rest of the proof is similar to that of Theorem . in Dung and Tien [] and is

omit-ted.

Proof of Theorem. We use the same notation as that in the proof of Theorem .. Taking q=  anda=in Lemma ., by (.) and Lemma ., it follows that

n=

n–E

max

≤j≤n|Sj|–εn

α+

C

n=

nα–E

max

≤j≤n

j

i=

(Xni–EXni)

+ ∞

n=

n–E

max

≤j≤n

j

i=

XniEXni

C

n=

nα–

n

i=

EXiI |Xi| ≤

+nαP |X

i|>

+ ∞

n=

n– n

i=

E|Xi|I |Xi|>

C

n=

nα–EXI |X| ≤nα+C

n=

n–E|X|I |X|>nα

C

n=

nα– n

m=

mαP m–  <|X|/αm

+C

n=

n– ∞

m=n

E|X|I m<|X|/αm+ 

=C

m=

mαP m–  <|X|/αm

n=m nα–

+C

m=

E|X|I m<|X|/αm+  m

n=

n–

C

m=

P m–  <|X|/αm+C

m=

E|X|I m<|X|/αm+ ln+mCE|X|+CE|X|ln+|X|<∞.

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Proof of Theorem. Inspired by the proof of Theorem . of Gut [], we have that

n=

nαp–P

sup j≥n

Sjjα

ε

= ∞

m= m–

n=m–

nαp–P

sup j≥n

Sjjα

ε

C

m=

P

sup j≥m–

Sjjα

ε

m–

n=m–

m(αp–)

C

m=

m(αp–)P

sup j≥m–

Sj

ε

=C

m=

m(αp–)P

sup k≥m

max

k–≤j<k

jSjα

ε

C

m=

m(αp–) ∞

k=m P

max

≤j≤k|Sj| ≥ε

α(k–)

C

k=

P

max

≤j≤k|Sj| ≥ε

α(k–)

k

m=

m(αp–)

C

k=

k(αp–)P

max

≤j≤k|Sj| ≥ε

α(k–)

C

k= k+–

n=k nαp–P

max

≤j≤n|Sj| ≥

εα

C

n=

nαp–P

max

≤j≤n|Sj| ≥

εα

. (.)

The desired result (.) follows from (.) and (.) immediately.

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 Andrei Volodin and anonymous referees for careful reading of the manuscript and valuable suggestions which helped in signiﬁcantly improving an earlier version of this paper. The research was supported by the National Natural Science Foundation of China (11171001, 11201001, 11126176), Natural Science Foundation of Anhui Province (1208085QA03), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Academic Innovation Team of Anhui University (KJTD001B), Doctoral Research Start-up Funds Projects of Anhui University, and the Talents Youth Fund of Anhui Province Universities (2011SQRL012ZD).

Received: 14 July 2012 Accepted: 1 October 2012 Published: 12 October 2012 References

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doi:10.1186/1029-242X-2012-229

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