A note on the strong limit theorem for weighted sums of sequences of negatively dependent random variables

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R E S E A R C H

Open Access

A note on the strong limit theorem for

weighted sums of sequences of negatively

dependent random variables

Haiwu Huang

1,2

_{and Dingcheng Wang}

1,3*

*_{Correspondence:}

[email protected]

1_{School of Mathematics Science,}

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

3_{Institute of Financial Engineering,}

School of Finance, School of Applied Mathematics, Nanjing Audit University, Nanjing, 211815, P.R. China

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

Abstract

In this paper, the strong limit theorem for weighted sums of sequences of negatively dependent random variables is further studied. As an application, the complete convergence theorem for sequences of negatively dependent random variables is obtained. Our results partly generalize and improve the corresponding results of Cai (Metrika 68:323-331, 2008) and Wanget al.(Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. a Mat., 2011, doi:10.1007/s13398-011-0048-0) to negatively dependent random variables under mild moment conditions.

MSC: 60F15

Keywords: negatively dependent random variables; complete convergence; weighted sums

1 Introduction

Deﬁnition . The random variablesX,X, . . . ,Xn are said to be negatively dependent

(ND) if

P(X≤x,X≤x, . . . ,Xn≤xn)≤ n

j=

P(Xj≤xj);

and

P(X>x,X>x, . . . ,Xn>xn)≤ n

j=

P(Xj>xj),

for allx,x, . . . ,xn∈R. An inﬁnite sequence of random variables{Xi;i≥}is said to be

ND if every ﬁnite subsetX,X, . . . ,Xnis ND.

An array of random variables{Xni;i≥,n≥}is called rowwise ND random variables

if for everyn≥,{Xni;i≥}is a sequence of ND random variables.

Deﬁnition . The random variablesX,X, . . . ,Xn,n≥ are said to be negatively

asso-ciated (NA) if for every pair of disjoint subsetsAandAof{, , . . . ,n},

covf(Xi;i∈A),f(Xj;j∈A)

≤,

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wheneverfandfare increasing for every variable (or decreasing for every variable), such that the covariance exists. An inﬁnite family{Xi;i≥}of random variables is said to be

NA if every ﬁnite subfamily is NA.

The concept of ND random variables was introduced by Ebrahimi and Ghosh[], and the concept of NA random variables was introduced by Joag-Dev and Proschan []. Obviously, independent random variables are ND. Joag-Dev and Proschan [] pointed out that NA random variables are ND. They also presented an example in whichX= (X,X,X,X) possesses ND, but does not possess NA. So, we can see that ND is much weaker than NA. Because of the wide applications of ND random variables, the notions of ND random vari-ables have received more and more attention recently. A large number of limit theorems for ND random variables have been established by many authors. We can refer to [–]

etc.Hence, extending the limit properties of independent or NA random variables to the case of ND random variables is highly desirable and of considerable signiﬁcance in theory and application.

As Bai and Cheng [] remarked, many useful linear statistics based on a random sample are weighted sums of independent identically distributed (i.i.d.) random variables. Exam-ples include least-squares estimators, nonparametric regression function estimators, jack-knife estimates and so on. In this respect, studies of strong laws for these weighted sums have demonstrated signiﬁcant progress in probability theory with applications in math-ematical statistics; many authors studied the strong laws for weighted sums of random variables. In the case of independence, Bai and Cheng [] proved the following strong laws of large numbers for weighted sums.

Theorem .(Bai and Cheng []) Let{X,Xn;n≥}be a sequence of i.i.d.random vari-ables with EXn= . Suppose that <α,β <∞, ≤p< and/p= /α+ /β,and let {ani; ≤i≤n,n≥}be an array of real constants such that

Aα=lim sup

n→∞

Aα,n<∞, Aαα,n=n– n

i=

|ani|α. (.)

If E|Xn|β_<_∞_,_then

lim

n→∞n

–/p

_n

i=

aniXi

= , a.s. (.)

Theorem .(Bai and Cheng []) Let{X,Xn;n≥}be a sequence of i.i.d.random vari-ables and

Eexph|X|γ <∞ for some h> and someγ> , (.)

and let{ani; ≤i≤n,n≥}be an array of real constants such that(.)satisﬁes forα∈

(, ).Then for <α≤and bn=n/α(logn)/γ,

n

i=

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Moreover,for <α< and bn=n/α(logn)/γ+δ,and EXn= ,

n

i=

aniXi/bn→ a.s., (.)

whereδ=γ(α– )/α( +γ).

Wu[]generalized and improved the above Theorem.to ND random variables, re-moved the identically distributed condition as follows.

Theorem .(Wu []) Let{Xn;n≥}be a sequence of ND random variables which is stochastically dominated by a random variable X.Suppose that <α,β <∞,  <p< 

and/p= /α+ /β.Ifβ> ,further assume that EXn= .Let{ani; ≤i≤n,n≥}be an array of real constants such that

n

i=

|ani|αn. (.)

If E|Xn|β_<_∞_,_then

lim

n→∞n –/p

_n

i=

aniXi

= , a.s. (.)

Recently,Cai[]and Wang et al.[]have studied the complete convergence for NA ran-dom variables and arrays of rowwise ND ranran-dom variables under the exponential moment conditions respectively.Their results generalize and improve the above Theorem.to NA and ND random variables.Inspired by Cai[],Wang et al.[],and other papers men-tioned above,we investigate the limit behavior of ND random variables and obtain some complete convergence results.We use methods diﬀerent from those of Cai[]and Wang et al.[].

The main purpose of this paper is to further study the complete convergence for ND random variables and arrays of rowwise ND random variables under weaker moment conditions. As applications, the complete convergence for linear statistics of ND random variables and arrays of rowwise ND random variables are obtained without assumptions of being identically distributed. The results obtained not only generalize the above The-orem . to the case of ND and arrays of rowwise ND random variables, but also partly improve the corresponding results of Cai [] and Wanget al.[].

Throughout this paper,Cwill represent a positive constant whose value may change from one appearance to the next, andan=O(bn) will meanan≤C(bn).

We will use the following concept in this paper. Let{Xn;n≥}be a sequence of random

variables, and letXbe a nonnegative random variable. If there exists a constantC( <C<

∞) such that

P|Xn| ≥t≤CP(X≥t), (.)

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2 Main results and proofs

Now, we state and prove our main results of this paper.

Theorem . Let{Xn;n≥}be a sequence of ND random variables which is stochastically dominated by a random variable X.Let Tn=

n

i=aniXi,n≥,where the weights{ani: ≤ i≤n,n≥}are a triangular array of real constants such that ani= for i>n.Let

Aβ=lim sup

n→∞ Aβ,n<∞; Aβ,n=n –

n

i=

|ani|β_, _(.)

whereβ=max(α,γ)for some <α≤,γ> andα=γ.Assume that EXn= for <α≤ and E|X|β<∞.Then

∞

n=

n–P|Tn|>εbn

<∞ for∀ε> , (.)

where bn=n/α(logn)/γ.

In order to prove our results, we need the following lemmas.

Lemma .(Bozorgniaet al.[]) Let random variables X,X, . . . ,Xnbe ND,f,f, . . . ,fnbe all nondecreasing(or all nonincreasing)functions,then random variables f(X),f(X), . . . ,

fn(Xn)are ND.

Lemma . (Asadianet al.[]) Let{Xn;n≥} be a sequence of ND random variables with EXn= and E|Xn|p<∞for some p≥and for all n≥.Then there exists a positive constant C=C(p)depending only on p such that for all n≥,

E

n

i=

Xi

p

≤C

_n

i=

E|Xi|p₊

_n

i=

EX

i

p/

. (.)

Lemma . Let X be a random variable and{ani: ≤i≤n,n≥}be an array of constants satisfying(.),bn=n/α(logn)/γ.Then

∞

n=

n– n

i=

P|aniX|>bn

≤CE|X|β. (.)

The proof is similar to that of Lemma.of Sung[].So,we omit it.

Lemma . Let{Xn;n≥}be a sequence of random variables which is stochastically dom-inated by a random variable X.For any u> ,t> and n≥,the following two statements hold:

E|Xn|uI|Xn| ≤t≤CE|X|uI|X| ≤t+tuP|X|>t; (.)

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Proof of Theorem. Without loss of generality, assume thatani≥ for all ≤i≤n,n≥.

Deﬁne that

X_i(n)=aniXiI

|aniXi| ≤bn

+bnI(aniXi>bn) –bnI(aniXi< –bn);

T_j(n)=

j

i=

X_i(n)–EX_i(n), for alli≥.

By Lemma ., we can see that for ﬁxedn≥,{X(_in);i≥}is still a sequence of ND random variables. It is easy to check that for anyε> ,

n i=

aniXi

>εbn

⊂max

≤i≤n|aniXi|>bn

∪ n

i=

X(_in)

>εbn

,

which implies that

P|Tn|>εbn

≤P

max

≤i≤n|aniXi|>bn

+P

Tn(n)+

n

i=

EX(_in)

>εbn

≤ n

i=

P|aniXi|>bn

+P

T_n(n)>εbn–

n i=

EX_i(n)

. (.)

Firstly, we will show that

b–_n

n i=

EX_i(n)

→, asn→ ∞. (.)

Ifγ >α, byn_i_=|ani|β=

n i=|ani|

γ_≤_Cn_{and Lyapunov’s inequality,}



n n

i=

|ani|α≤

 n n i=

|ani|γ

α/γ

≤C,

which implies that _n×n×max_≤_i_≤_n|ani|α₌_max

≤i≤n|ani|α≤Cfor∀n≥.

Ifγ <α, it easily follows thatmax_≤i≤n|ani|α≤Cfor∀n≥.

For  <α≤, it follows from Lemma . andE|X|β_<_∞_that

b–_n

n i=

EX_i(n)

≤b–_n n

i= EX(_in)

≤b–_n n

i=

E|aniXi|I

|aniXi| ≤bn

+

n

i=

P|aniXi|>bn

≤Cb–_n n

i=

E|aniX|I

|aniX| ≤bn

+bnP

|aniX|>bn

+CnP|aniX|>bn

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≤Cb–_n n

i=

E|aniX|I

|aniX| ≤bn

+CnP|aniX|>bn

=C n i= E _|

aniX| bn

I|aniX| ≤bn

+CnP|aniX|>bn

≤C n i= E _|

aniX| bn

α

I|aniX| ≤bn

+CnP|aniX|>bn

≤C n i= E

|aniX| bn

α

+CnP|aniX|>bn

=Cb–_nα n

i=

|ani|αE|X|α+CnP|aniX|>bn

≤C(logn)–α/γE|X|α₊_CnE|aniX| α

bα

n

≤C(logn)–α/γE|X|α+C(logn)–α/γ|ani|αE|X|α→ asn→ ∞. (.)

For  <α≤, it follows from Lemma .,EXn=  andE|X|β<∞that

b–_n

n i=

EX_i(n)

≤b–_n n

i= EX(_in)

≤b–_n n

i=

E|aniXi|I

|aniXi| ≤bn

+

n

i=

P|aniXi|>bn

≤Cb–_n n

i=

E|aniX|I

|aniX| ≤bn

+bnP

|aniX|>bn

+CnP|aniX|>bn

≤Cb–_n n

i=

E|aniX|I

|aniX|>bn

+CnP|aniX|>bn

≤C n i= E _|

aniX| bn

α

I|aniX|>bn

+CnP|aniX|>bn

≤C n i= E

|aniX| bn

α

+CnP|aniX|>bn

≤C(logn)–α/γE|X|α₊_C₍_log_n₎–α/γ_|

ani|α_E_|_X_|α_→_, _as_n_{→ ∞}_. _(.)

From (.) and (.), we can get (.) immediately. Hence, fornlarge enough,

P|Tn|>εbn

≤ n

i=

P|aniXi|>bn

+PT_n(n)>εbn 

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Secondly, we need only to prove that

I

∞

n=

n– n

i=

P|aniXi|>bn

<∞; (.)

II

∞

n=

n–PT_n(n)>εbn 

<∞. (.)

It follows from Lemma . and Lemma . that

I

∞

n=

n– n

i=

P|aniXi|>bn

≤C

∞

n=

n– n

i=

P|aniX|>bn

≤CE|X|β<∞, (.)

whereβ=max(α,γ) for some  <α≤ andγ > .

It follows from Lemma . and the Markov inequality that

II

∞

n=

n–PT(n)

n >

εbn



≤C

∞

n=

n–b–_npET_n(n)p

≤C

∞

n=

n–b–_np

_n

i=

EX_i(n)p+ _n

i=

EX_i(n)

p/

II+II. (.)

Let p≥max{,γ}, α <γ, note that E|X|β ₌_E_|_X_|γ _<_∞_{. It follows from Lemma .,}

Lemma . and the Markov inequality that

II =C ∞

n=

n–b–_np n

i=

EX_i(n)p

≤C

∞

n=

n–b–_np

_n

i=

|ani|pE|Xi|pI

|aniXi| ≤bn

+

n

i=

bp_nP|aniXi|>bn

≤C

∞

n=

n–b–_np

_n

i=

|ani|pE|X|pI

|aniX| ≤bn

+

n

i=

bp_nP|aniX|>bn

≤C

∞

n=

n–b–_nγ n

i=

|ani|γE|X|γ +C

∞

n=

n– n

i=

P|aniX|>bn

≤C

∞

n=

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Ifα>γ, note thatE|X|β=E|X|α<∞. We can get that

II =C ∞

n=

n–b–_np n

i=

EX_i(n)p

≤C

∞

n=

n–b–_np

_n

i=

|ani|pE|Xi|pI

|aniXi| ≤bn

+

n

i=

bp_nP|aniXi|>bn

≤C

∞

n=

n–b–_nα n

i=

|ani|αE|X|α+C

∞ n= n– n i=

P|aniX|>bn

≤C

∞

n=

n–(logn)–α/γ +CE|X|α<∞. (.)

Next, we will proveII=C ∞

n=n–b –p n (

n i=E|X

(n)

i |



)p/<∞in the following two cases. Ifα<γ ≤ orγ <α≤, letp≥max{,_αγ}. Noting thatE|X|α_<_∞_{, we can get that}

II =C ∞

n=

n–b–_np

_n

i=

EX_i(n)

p/

≤C

∞

n=

n–b–_np

_n

i=

E|aniXi|I

|aniXi| ≤bn

p/

+C

∞

n=

n–b–_np

_n

i=

|bn|P|aniXi|>bn

p/

≤C

∞

n=

n–b–_np

_n

i=

E|aniX|I

|aniX| ≤bn

+b_nP|aniX|>bn

p/ +C ∞ n= n– _n i=

P|aniX|>bn

p/ ≤C ∞ n= n– _n i=

E|aniX|



b

n

I|aniX| ≤bn

p/ +C ∞ n= n– _n i=

P|aniX|>bn

p/ ≤C ∞ n= n– _n i=

E|aniX|

α

bα

n

I|aniX| ≤bn

p/ +C ∞ n= n– _n i=

E|aniX|

α bα n p/ ≤C ∞ n=

n–b–_nαp/

_n

i=

|ani|αE|X|α

p/

≤C

∞

n=

n–(logn)–αp/(γ)<∞. (.)

Ifγ > ≥αorγ ≥ >α, byn_i_=|ani|β₌n i=|ani|

γ _≤_Cn_{and Lyapunov’s inequality,}

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we can obtain that



n n

i=

|ani|α≤



n n

i=

|ani|γ

α/γ

≤C,

which implies thatn_i_=|ani|α=O(n). Hence, frommax≤i≤n|ani|α≤Cnand the Hölder

inequality, then∀k≥α,

n

i=

|ani|k=

n

i=

|ani|α_|_a

ni|k–α≤Cnn k–α

α ≤_Cnkα_.

So,

n

i=

|ani|=On/α.

Letp>γ,

II =C ∞

n=

n–b–_np

_n

i=

EX(_in)

p/

≤C

∞

n=

n–b–_np

_n

i=

|ani|

p/ +C

∞

n=

n–b–_np

_n

i=

|ani|α

p/

≤C

∞

n=

n–b–_npnp/α

≤C

∞

n=

n–(logn)–p/γ <∞. (.)

Hence, the desired result (.) follows from (.)-(.) immediately. The proof of

Theo-rem . is complete.

Similar to the proof of Theorem ., we can get the following results for arrays of rowwise ND random variables.

Theorem . Let{Xni;i≥,n≥}be an array of rowwise ND random variables which is stochastically dominated by a random variable X.Let Tn=

n

i=aniXni,n≥,where the weights{ani: ≤i≤n,n≥}are a triangular array of real constants such that(.).Let bn=n/α(logn)/γ.If EXni= for <α≤and E|X|β<∞,then(.)holds true.

Remark . Note that the results of Cai [] and Wanget al.[] provide a stronger con-clusion on the complete convergence for maximums of partial sums under the exponential moment condition than the results presented in Theorem . and Theorem . above for α=γ; that is, they obtained results of the form

∞

n=

n–P

max

≤j≤n|Tj|>εbn

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It is still an open problem to obtain results of this type for ND random variables under the conditions ofα=γ andα=γ. One suggests that a solution can be obtained if a better moment inequality than that presented above in Lemma . could be established.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

DW participated in the design of the study. HH conceived of the study, and participated in the design and proof of the article. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu, 610054, P.R. China.} 2_{College of Science, Guilin University of Technology, Guilin, 541004, P.R. China.}3_{Institute of Financial Engineering, School}

of Finance, School of Applied Mathematics, Nanjing Audit University, Nanjing, 211815, P.R. China.

Acknowledgements

The authors are very grateful to the anonymous referees and the editor Prof Andrei Volodin for their valuable comments and some helpful suggestions that improved the clarity and readability of the article. This work was supported by the Project Supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011]47), the National Natural Science Foundation of China (No:71271042, 11061012), the Plan of Jiangsu Specially-Appointed Professors and the Major Program of Key Research Center in Financial Risk Management of Jiangsu Universities of philosophy and social sciences (No:2012JDXM009).

Received: 15 February 2012 Accepted: 10 August 2012 Published: 17 October 2012 References

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