DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2016/9039345 9039345 Research Article Complete Moment Convergence for Negatively Dependent Sequences of Random Variables http://orcid.org/0000-0001-6673-9762 Wu Qunying 1 http://orcid.org/0000-0001-6578-6225 Jiang Yuanying 1 Cordero Alicia College of Science Guilin University of Technology Guilin 541004 China glut.edu.cn 2016 2772016 2016 25 05 2016 28 06 2016 2016 Copyright © 2016 Qunying Wu and Yuanying Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the complete moment convergence for sequences of negatively dependent identically distributed random variables with EX=0,  Eexp|X|α<,  0<α<1, and Eexp|X|ln-r|X|<,  r>0. As a result, we establish the new complete moment convergence theorems.

National Natural Science Foundation of China 11361019 Support Program of the Guangxi China Science Foundation 2015GXNSFAA139008
1. Introduction and Main Results

Random variables X and Y are said to be negative quadrant dependent (NQD) if(1)PXx,YyPXxPYyfor all x,yR. A collection of random variables is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the collection satisfies (1).

It is important to note that (1) implies(2)PX>x,Y>yPX>xPY>yfor all x,yR. Moreover, it follows that (2) implies (1) and, hence, (1) and (2) are equivalent. However, Ebrahimi and Ghosh  showed that (1) and (2) are not equivalent for a collection of 3 or more random variables. Accordingly, the following definition is needed to define sequences of negatively dependent random variables.

Definition 1.

Random variables X1,,Xn are said to be negatively dependent (ND) if, for all real x1,,xn, (3)Pj=1nXjxjj=1nPXjxj,Pj=1nXj>xjj=1nPXj>xj.An infinite sequence of random variables {Xn;n1} is said to be ND if every finite subset X1,,Xn is ND.

Definition 2.

Random variables X1,X2,,Xn,n2, are said to be negatively associated (NA) if, for every pair of disjoint subsets A1 and A2 of {1,2,,n}, (4)covf1Xi;iA1,f2Xj;jA20, where f1 and f2 are increasing for every variable (or decreasing for every variable) function such that this covariance exists. A sequence of random variables {Xi;i1} is said to be NA if its every finite subfamily is NA.

The definition of PNQD is given by Lehmann . The definition of NA is introduced by Joag-Dev and Proschan , and the concept of ND is given by Bozorgnia et al. . These concepts of dependent random variables are very useful for reliability theory and applications.

It is easy to see from the definitions that NA implies ND. But Example 1.5 in Wu and Jiang  shows that ND does not imply NA. Thus, it is shown that ND is much weaker than NA. In the articles listed earlier, a number of well-known multivariate distributions are shown to possess the ND properties. In many statistics and mechanic models, a ND assumption among the random variables in the models is more reasonable than an independent or NA assumption. Because of wide applications in multivariate statistical analysis and reliability theory, the notions of ND random variables have attracted more and more attention recently. A series of useful results have been established (cf. Bozorgnia et al. , Fakoor and Azarnoosh , Asadian et al. , Wu [5, 8], Wang et al. , and Liu et al. ). Hence, it is highly desirable and of considerable significance in the theory and application to study the limit properties of ND random variables theorems and applications.

Chow  first investigated the complete moment convergence, which is more exact than complete convergence. Thus, complete moment convergence is one of the most important problems in probability theory. The recent results can be found in Chen and Wang , Gut and Stadtmüller , Sung , Guo , and Qiu and Chen [16, 17]. In addition, Qiu and Chen  obtained complete moment convergence theorems for independent identically distributed sequences of random variables with EX=0,  Eexplnα|X|<, α>1. A natural question is whether there is any type of complete moment convergence theorems for 0<α1. In this paper, we study the complete moment convergence for sequences of negatively dependent identically distributed random variables with EX=0,  Eexp|X|α<,  0<α<1, and Eexp|X|ln-r|X|<,  r>0. As a result, we establish the new complete moment convergence theorems.

In the following, the symbol c stands for a generic positive constant which may differ from one place to another. Let anbn denote that there exists a constant c>0 such that ancbn for sufficiently large n, lnx mean ln(max(x,e)), and I denotes an indicator function.

Theorem 3.

Let 0<α<1, {X,Xn;n1} be a sequence of ND identically distributed random variables with partial sums Sn=i=1nXi,  n1. Suppose that(5)EX=0,EexpXα<,and then(6)n=1expnαnpEmax1knSk-βn+q<β>1,q>0  and  all  pR.Conversely, if (6) holds for p=α-2-q and some β>0, then(7)EexpX4βα<.

For α=1, we have the following.

Theorem 4.

Let {X,Xn;n1} be a sequence of ND identically distributed random variables with partial sums Sn=i=1nXi,  n1. Suppose that(8)EX=0,EexpXln-rX<for  some  r>0,and then(9)n=1expnln-rnnpEmax1knSk-βn+q<β>1,q>0  and  all  pR.Conversely, if n=1expnln-rnn-q-1ln-rnEmax1kn|Sk|-βn+q< for some β>0, then(10)EexpX4βln-rX4β<.

Remark 5.

By mimicking the analogous part in the proof of Theorem 2.1 in Qiu and Chen , (6) and (9) imply, respectively, (11)n=1expnαnpEsup1knSkk-β+q<β>1,q>0  and  all  pR,(12)n=1expnln-rnnpEsup1knSkk-β+q<β>1,q>0  and  all  pR.

Remark 6.

Because (6) and (9) hold for all q>0, pR, and ND random sequences, (6) and (9) are very broad conclusions.

2. Proofs

The following four lemmas play important roles in the proof of our theorems.

Lemma 7 (Bozorgnia et al. [<xref ref-type="bibr" rid="B2">4</xref>]).

Let {Xn;n1} be a sequence of ND random variables.

Let {fn;n1} be a sequence of Borel functions; all of them are monotone increasing (or all are monotone decreasing). Then, {fn(Xn);n1} is a sequence of ND r.v.’s.

Let X1,,Xn be nonnegative. Then, (13)Ej=1nXjj=1nEXj.

In particular, let t1,,tn be all nonnegative (or nonpositive) real numbers. Then, (14)Eexpj=1ntjXjj=1nEexptjXj.

Lemma 8.

Let {X,Xi;i1} be a sequence of ND identically distributed random variables with Eexp(λX)< for any λ>0. Assume that {ai;i1} is a sequence of positive real numbers such that an as n. Then, for Sk=i=1kXi,  λ>0, and a positive integer n, (15)Pmax1knSk>anexp-λank=1nEexpλX-EXk.

Proof.

Obviously, δ0=^sup{λ0;Eexp(λ(X-EX))<}= from condition Eexp(λX)< for any λ>0. By Lemma 7(ii), (1.1) in Wang et al.  holds for any λR. Together with condition 0<an, we know that the conditions of Theorem 2.2 in Wang et al.  are satisfied. Therefore, by Theorem 2.2 in Wang et al. , for any λ>0, (16)Pmax1knSk>anminλ0,exp-λank=1nEexpλX-EXkexp-λank=1nEexpλX-EXk.

Lemma 9.

For any random variable X,(17)EexpXα<n=1expnαn1-αPX>n<for  any  α>0,(18)EexpXln-rX<n=2expnln-rnln-rnPX>n<for  any  r>0.

Proof.

Let anbn denote that there exist constants c1>0 and c2>0 such that c1anbnc2an for sufficiently large n. We have (19)n=1expnαn1-αPX>n=n=1expnαn1-αj=nPj<Xj+1=j=1Pj<Xj+1n=1jexpnαn1-αj=1expjαEIj<Xj+1j=1EexpXαIj<Xj+1EexpXα, and it follows that (17) holds.

Note that (20)n=2jexpnln-rnln-rn2jexpxln-rxln-rxdx~2jexpxln-rxln-rx-rln-r-1xdx=2jexpxln-rxdxexpjln-rj, and hence, using similar methods used to prove (17), we can prove that (18) holds.

Lemma 10.

Let {Xn;n1} be a sequence of ND random variables. Then, for any x0, there exists a positive constant c such that, for all n1, (21)1-Pmax1knXk>x2k=1nPXk>xcPmax1knXk>x.

Further, if P(max1kn|Xk|>x)0 as n, then there exists a positive constant c such that, for all n1, (22)k=1nPXk>xcPmax1knXk>x.

Proof.

Obviously, ND implies pairwise negative quadrant dependent (PNQD) from the definitions of ND and PNQD. Thus, Lemma 10 holds from Lemma 1.4 of Wu .

Proof of Theorem <xref ref-type="statement" rid="thm1.3">3</xref>.

Note that(23)Emax1knSk-βn+q=βq0nqxq-1Pmax1knSk-βn>βxdx+βqnqxq-1Pmax1knSk-βn>βxdxnqPmax1knSk>βn+nxq-1Pmax1knSk>βxdx.Hence, in order to establish (6), it suffices to prove, for any pR,(24)n=1expnαnpPmax1knSk>βn<,(25)n=1expnαnpnxq-1Pmax1knSk>βxdx<.Firstly, we prove (24). Let β>1 be arbitrary; define, for 1kn, (26)X=-βnIX<-βn+XIXβn+βnIX>βn,Xk=-βnIXk<-βn+XkIXkβn+βnIXk>βn,Sn=k=1nXk.It is easy to get(27)Pmax1knSk>nβPmax1knSk>nβ+nPX>nβ.From (5) and the Markov inequality,(28)PX>nβexp-βαnαEexpXαexp-βαnα.

Set X~=X-EX; then EX~=0; using the obvious inequality ey1+y+(1+eyI(y>0))y2/2, it follows that, for any λ>0,(29)EexpλX~1+λ22EX~21+expλX~IX~>01+cλ2EX2+EX2expλX~.Let β-α<δ<δ1<1 and λ=δ(βn)α-1; then by cr inequality and E|X|E|X|<,(30)λX~λX+EXαX+EX1-αλXα+cX1-α+cδβnα-1Xα+cβn1-α+c=δXα+c1+o1δ1Xα+c.

By (5), EX2Eexp(|X|α)<, and EX2exp(δ1|X|α)Eexp(|X|α)<, therefore, by combination with (29) and (30), we get(31)EexpλX~1+cλ2EX2+cEX2expδ1Xα1+cλ2=1+cn2α-1expcn2α-1.

Obviously, Xk is increasing on Xk; thus, by Lemma 7(i), {Xk;k1} is also a sequence of ND random variables. Taking an=βn and λ=δ(βn)α-1 in Lemma 8, for 1<β1<δβα, we obtain(32)Pmax1knSk>nβexp-λβnk=1nEexpλX~knexp-λβn+cn2α-1=nexp-δβα+cn-1-αnαnexp-β1nα.

Replacing Xk by -Xk and by the same argument as above, (33)Pmax1kn-Sk>nβnexp-β1nα also holds. Hence,(34)Pmax1knSk>nβ2nexp-β1nα.

Thus, by combination with (27) and (28) and the fact that n=1(np/expc1nα)< for all c1>0  pR, we obtain that, for β2=^min(βα,β1)>1, (35)n=1expnαnpPmax1knSk>nβn=1np+1expβ2-1nα<.

Hence, (24) holds. Next, we prove (25).

Let xn. Replace n by x in X and Xk. Using similar methods to those used in the proof of (27)–(34), there is β2>1 such that(36)Pmax1knSk>xβnexp-β2xα.Thus,(37)nxq-1Pmax1knSk>xβdxnnxq-1exp-β2xαdx~nnxq-1-q-αβ2α-1xq-1-αexp-β2xαdx=nnxq-αexp-β2xα-β2αdxnq-α+1exp-β2nα, and it implies that (38)n=1expnαnpnxq-1Pmax1knSk>xβdxn=1np+q+1-αexpβ2-1nα<.

That is, (25) holds.

Conversely, if (6) holds for p=α-q-2 and some β>0, then, by Emax1kn|Sk|-βn+qβq0nqxq-1Pmax1kn|Sk|-βn>βxdxβqnqPmax1kn|Sk|>2βn, we have (39)n=1expnαnα-2Pmax1knSk>2βn<;by combination with max1kn|Xk|2max1knSk, it follows that(40)n=1expnαn2-αPmax1knXk>4βn<,and it implies that P(max1kn|Xk|>4βn)0,  n. Thus, by Lemma 10, there is c>0 such that (41)nPX>4βncPmax1knXk>4βn. Consequently, by (40), (42)n=1expnαn1-αPX4β>n<, and, hence, Eexp|X/(4β)|α< from Lemma 9. This completes the proof of Theorem 3.

Proof of Theorem <xref ref-type="statement" rid="thm1.4">4</xref>.

Let X and X~ be defined as Theorem 3. Let β-1<δ<1, and λ=δln-r(βn); then, (43)λX~δln-rβnX+cδln-rXX+cδXln-rX+c from |X|βn and xln-rx being monotonically increasing on x.

Thus, by (8), EX2exp(δ|X|ln-r|X|)Eexp(|X|ln-r|X|)<; therefore, similar to (29), we have (44)EexpλX~1+cln-2rnexpcln-2rn.

By Lemma 8, for 1<β1<δβ<β, we get(45)Pmax1knSk>nβnexp-λβn+cnln-2rn=nexp-δβnln-rβn+cnln-2rnnexp-β1nln-rn.

Replacing Xk by -Xk and by the same argument as above, (46)Pmax1kn-Sk>nβnexp-β1nln-rn also holds. Hence, (47)Pmax1knSk>nβ2nexp-β1nln-rn.

On the other hand, from (8) and the Markov inequality, (48)PX>nβexp-βnln-rβnexp-β1nln-rn.

Hence, together with (27),(49)n=1expnln-rnnpPmax1knSk>nβn=1np+1expβ1-1nln-rn<.

Similar to (36) and the above discussion, for xn, (50)Pmax1knSk>xβnexp-β1xln-rx. Thus,(51)nxq-1Pmax1knSk>xβdxnnxq-1exp-β1xln-rxdx~nnxq-1-rxq-1ln-1x-q-1β1-1xq-2lnrx-rβ1-1xq-2lnr-1xexp-β1xln-rxdx=nnxq-1lnrxexp-β1xln-rx-β1dxnqlnrnexp-β1nln-rn, and it implies that (52)n=1expnln-rnnpnxq-1Pmax1knSk>xβdxn=1np+qlnrnexpβ1-1nln-rn<. Thus, by combination with (23) and (49), (9) holds.

Conversely, by (18), using similar methods to those used in the proof of (7), we can get (10). This completes the proof of Theorem 4.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

Qunying Wu conceived the study and drafted, completed, read, and approved the final paper. Yuanying Jiang conceived the study and completed, read, and approved the final paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11361019) and the Support Program of the Guangxi China Science Foundation (2015GXNSFAA139008).

Ebrahimi N. Ghosh M. Multivariate negative dependence Communications in Statistics —Theory and Methods 1981 10 4 307 337 10.1080/03610928108828041 MR612400 Lehmann E. L. Some concepts of dependence Annals of Mathematical Statistics 1966 37 5 1137 1153 10.1214/aoms/1177699260 MR0202228 Joag-Dev K. Proschan F. Negative association of random variables with applications The Annals of Statistics 1983 11 1 286 295 10.1214/aos/1176346079 MR684886 Bozorgnia A. Patterson R. F. Taylor R. L. Limit theorems for ND r.v.'s 1993 Athens, Greece University of Georgia Wu Q. Y. Jiang Y. Y. The strong consistency of M estimator in a linear model for negatively dependent random samples Communications in Statistics. Theory and Methods 2011 40 3 467 491 10.1080/03610920903427792 MR2765842 2-s2.0-78649277373 Fakoor V. Azarnoosh H. A. Probability inequalities for sums of negatively dependent random variables Pakistan Journal of Statistics 2005 21 3 257 264 MR2206118 ZBL1129.60303 Asadian N. Fakoor V. Bozorgnia A. Rosen-thal's type inequalities for negatively orthant dependent random variables Journal of the Iranian Chemical Society 2006 5 1-2 66 75 Wu Q. Y. Complete convergence for negatively dependent sequences of random variables Journal of Inequalities and Applications 2010 2010 10 507293 10.1155/2010/507293 MR2611036 2-s2.0-77952557726 Wang X. J. Deng X. Zheng L. L. Hu S. H. Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications Statistics 2014 48 4 834 850 10.1080/02331888.2013.800066 MR3234065 2-s2.0-84904729083 Liu C.-C. Guo M.-L. Zhu D.-J. Equivalent conditions of complete convergence for weighted sums of sequences of extended negatively dependent random variables Communications in Mathematical Research 2015 31 1 40 50 MR3363913 Chow Y. S. On the rate of moment convergence of sample sums and extremes Bulletin of the Institute of Mathematics. Academia Sinica 1988 16 3 177 201 MR1089491 ZBL0655.60028 Chen P. Y. Wang D. C. Complete moment convergence for sequence of identically distributed-mixing random variables Acta Mathematica Sinica 2010 26 4 679 690 10.1007/s10114-010-7625-6 MR2591647 2-s2.0-77952849939 Gut A. Stadtmüller U. An intermediate Baum-Katz theorem Statistics & Probability Letters 2011 81 10 1486 1492 10.1016/j.spl.2011.05.008 MR2818659 2-s2.0-79960045170 Sung S. H. Complete qth moment convergence for arrays of random variables Journal of Inequalities and Applications 2013 2013, article 24 10.1186/1029-242x-2013-24 2-s2.0-84874074236 Guo M. L. Equivalent conditions of complete moment convergence of weighted sums for ϕ-mixing sequence of random variables Communications in Statistics. Theory and Methods 2014 43 10–12 2527 2539 10.1080/03610926.2013.809116 MR3217830 2-s2.0-84902438315 Qiu D. H. Chen P. Y. Complete and complete moment convergence for weighted sums of widely orthant dependent random variables Acta Mathematica Sinica 2014 30 9 1539 1548 10.1007/s10114-014-3483-y MR3245935 2-s2.0-84905680120 Qiu D. H. Chen P. Y. Complete moment convergence for i.i.d. random variables Statistics & Probability Letters 2014 91 76 82 10.1016/j.spl.2014.04.001 MR3208119 2-s2.0-84899702268 Wang Y. B. Li Y. W. Gao Q. W. On the exponential inequality for acceptable random variables Journal of Inequalities and Applications 2011 2011, article 40 10 10.1186/1029-242x-2011-40 MR2837895 2-s2.0-84868106982 Wu Q. Y. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables Journal of Applied Mathematics 2012 2012 10 104390 MR2948134