AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 521618 10.1155/2013/521618 521618 Research Article The Strong Consistency of the Estimator of Fixed-Design Regression Model under Negatively Dependent Sequences Wang Xuejun Ge Meimei Hu Shuhe Wang Xize Bertsch Michiel 1 School of Mathematical Science Anhui University Hefei 230601 China ahu.edu.cn 2013 26 11 2013 2013 25 06 2013 07 10 2013 2013 Copyright © 2013 Xuejun Wang et al. 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 strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.

1. Introduction

Let {Xn,n1} be a sequence of random variables defined on a fixed probability space (Ω,,P). It is well known that the Rosenthal-type inequality for the partial sum i=1nXi plays an important role in probability limit theory and mathematical statistics. The main purpose of the paper is to investigate the strong consistency of the estimator of fixed design regression model under negatively dependent sequences, by using the Rosenthal-type inequality.

Consider the following fixed design regression model: (1)Yni=g(xni)+ɛni,i=1,2,,n,n1, where xni are known fixed design points from A, where Ap is a given compact set for some p1, g(·) is an unknown regression function defined on A, and ɛni are random errors. Assume that, for each n1, (ɛn1,ɛn2,,ɛnn) have the same distribution as (ɛ1,ɛ2,,ɛn). As an estimator of g(·), the following weighted regression estimator will be considered: (2)gn(x)=i=1nWni(x)Yni,xAp, where Wni(x)=Wni(x;xn1,xn2,,xnn), i=1,2,,n, are the weight function.

The above estimator was first proposed by Georgiev  and subsequently has been studied by many authors. For instance, when ɛni are assumed to be independent, consistency and asymptotic normality have been studied by Georgiev and Greblicki , Georgiev , and Müller  among others. Results for the case when ɛni are dependent have also been studied by various authors in recent years. Fan  extended the work of Georgiev  and Müller  in the estimation of the regression model to the case which form an Lq-mixingale sequence for some 1q2. Roussas  discussed strong consistency and quadratic mean consistency for gn(x) under mixing conditions. Roussas et al.  established asymptotic normality of gn(x) assuming that the errors are from a strictly stationary stochastic process and satisfying the strong mixing condition. Tran et al.  discussed again asymptotic normality of gn(x) assuming that the errors form a linear time series, more precisely, a weakly stationary linear process based on a martingale difference sequence. Hu et al.  studied the asymptotic normality for double array sum of linear time series. Hu et al.  gave the mean consistency, complete consistency, and asymptotic normality of regression models with linear process errors. Liang and Jing  presented some asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences; Yang et al.  generalized the results of Liang and Jing  for negatively associated sequences to the case of negatively orthant dependent sequences. Shen  presented the Bernstein-type inequality for widely dependent random variables and gave its application to nonparametric regression models, and so forth. The main purpose of this section is to investigate the strong consistency of estimator of the fixed design regression model based on negatively dependent random variables.

The concept of negatively dependent random variables was introduced by Lehmann  as follows.

A finite collection of random variables X1,X2,,Xn is said to be negatively dependent (or negatively orthant dependent, ND in short) if (3)P(X1>x1,X2>x2,,Xn>xn)i=1nP(Xi>xi),P(X1x1,X2x2,,Xnxn)i=1nP(Xixi) hold for all x1,x2,,xn. An infinite sequence {Xn,n1} is said to be ND if every finite subcollection is ND.

Obviously, independent random variables are ND. Joag-Dev and Proschan  pointed out that negatively associated (NA, in short) random variables are ND. They also presented an example in which X=(X1,X2,X3,X4) possesses ND but does not possess NA. The another example which is ND but is not NA was provided by Wu  as follows.

Example 1.

Let Xi be a binary random variable such that P(Xi=1)=P(Xi=0)=0.5 for i=1,2,3. Let (X1,X2,X3) take the values (0,0,1), (0,1,0), (1,0,0), and (1,1,1), each with probability 1/4. It can be verified that all the ND conditions hold. However, (4)P(X1+X31,X20)=48P(X1+X31)P(X20)=38. Hence, X1,X2, and X3 are not NA.

So we can see that ND is weaker than NA. A number of well-known multivariate distributions have the ND properties, such as multinomial, convolution of unlike multinomials, multivariate hypergeometric, dirichlet, dirichlet compound multinomial, and multinomials having certain covariance matrices. Because of the wide applications of ND random variables, the limiting behaviors of ND random variables have received more and more attention recently. A number of useful results for ND random variables have been established by many authors. We refer to Volodin  for the Kolmogorov exponential inequality, Asadian et al.  for Rosenthal's type inequality, Kim  for Hájek-Rényi type inequality, Amini et al. [20, 21], Ko and Kim , Klesov et al. , and Wang et al.  for almost sure convergence, Amini and Bozorgnia , Kuczmaszewska , Taylor et al. , Zarei and Jabbari , Wu [16, 29], Sung , and Wang et al.  for complete convergence, Wang et al.  for exponential inequalities and inverse moment, Shen  for strong limit theorems for arrays of rowwise ND random variables, Shen  for strong convergence rate for weighted sums of arrays of rowwise ND random variables, and so on. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.

This work is organized as follows: some preliminary lemmas are presented in Section 2, and the main results and their proofs are provided in Section 3.

Throughout the paper, C denotes a positive constant not depending on n, which may be different in various places. an=O(bn) represents anCbn for all n1. Let [x] denote the integer part of x, and let I(A) be the indicator function of the set A. Denote x+=xI(x0) and x-=-xI(x<0).

2. Preliminaries

In this section, we will present some important lemmas which will be used to prove the main results of the paper.

Lemma 2 (cf. Bozorgnia et al. [<xref ref-type="bibr" rid="B35">35</xref>]).

Let the random variables X1,X2,,Xn be ND, and let f1,f2,,fn be all nondecreasing (or all nonincreasing) functions. Then random variables f1(X1),f2(X2),,fn(Xn) are ND.

Lemma 3 (cf. Asadian et al. [<xref ref-type="bibr" rid="B18">18</xref>]).

Let p2, and let {Xn,n1} be a sequence of ND random variables with EXn=0 and E|Xn|p< for every n1. Then there exists a positive constant C depending only on p such that for every n1, (5)E|i=1nXi|pC{i=1nE|Xi|p+(i=1nEXi2)p/2}.

The following concept of stochastic domination will be used in this work.

Definition 4.

A sequence {Xn,n1} of random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C such that (6)P(|Xn|>x)CP(|X|>x) for all x0 and n1.

By the definition of stochastic domination and integration by parts, we can get the following property for stochastic domination. For the details of the proof, one can refer to Wu [36, 37] or Shen and Wu .

Lemma 5.

Let {Xn,n1} be a sequence of random variables which is stochastically dominated by a random variable X. For any α>0 and b>0, the following two statements hold: (7)E|Xn|αI(|Xn|b)C1[E|X|αI(|X|b)+bαP(|X|>b)],E|Xn|αI(|Xn|>b)C2E|X|αI(|X|>b), where C1 and C2 are positive constants.

3. Main Results and Their Proofs

Unless otherwise specified, we assume throughout the paper that gn(x) is defined by (2). For any function g(x), we use c(g) to denote all continuity points of the function g on A. The norm x is the Eucledean norm. For any fixed design point xA, the following assumptions on weight function Wni(x) will be used:

i=1nWni(x)1 as n;

i=1n|Wni(x)|C< for all n;

i=1n|Wni(x)|·|g(xni)-g(x)|I(xni-x>a)0 as n for all a>0.

Based on the assumptions above, we can get the following strong consistency of the fixed design regression estimator gn(x).

Theorem 6.

Let {ɛn,n1} be a sequence of mean zero ND random variables, which is stochastically dominated by a random variable X. Assume that conditions (A1)–(A3) hold true. If E|X|p< for some p>1, and if there exist some s(1/p,1) such that (8)max1in|Wni(x)|=O(n-s), then for any xc(g), (9)gn(x)g(x)a.s.,asn.

Proof.

For xc(g) and a>0, we have by (1) and (2) that (10)|Egn(x)-g(x)|i=1n|Wni(x)|·|g(xni)-g(x)|I(xni-xa)+i=1n|Wni(x)|·|g(xni)-g(x)|I(xni-x>a)+|g(x)|·|i=1nWni(x)-1|. Since xc(g), hence for any ɛ>0, there exist a δ>0 and x*-x<δ such that |g(x*)-g(x)|<ɛ. If we take a(0,δ) in (10), we can get that (11)|Egn(x)-g(x)|ɛi=1n|Wni(x)|+|g(x)|·|i=1nWni(x)-1|+i=1n|Wni(x)|·|g(xni)-g(x)|I(xni-x>a). Therefore, we have by conditions (A1)–(A3) that (12)limnEgn(x)=g(x),xc(g). For a fixed design point xc(g), without loss of generality, we assume that Wni(x)0 (otherwise, we use Wni+(x) and Wni-(x) instead of Wni(x), resp., and note that Wni(x)=Wni+(x)-Wni-(x)).

By (12), we can see that in order to prove (9), we only need to show that (13)gn(x)-Egn(x)=i=1nWni(x)ɛnii=1nRni0a.s.,asn, where Rni=Wni(x)ɛni.

For fixed ɛ>0, choose 1/p<δ<s and some positive integer N (to be specified later). Denote, for i=1,2,,n, (14)Xni(1)=-n-δI(Rni<-n-δ)+RniI(|Rni|n-δ)+n-δI(Rni>n-δ),Xni(2)=(Rni-n-δ)I(n-δ<Rni<ɛN),Xni(3)=(Rni+n-δ)I(-n-δ>Rni>-ɛN),Xni(4)=(Rni-n-δ)I(RniɛN)+(Rni+n-δ)I(Rni-ɛN). It is easy to check that Xni(1)+Xni(2)+Xni(3)+Xni(4)=Rni. Hence, to prove (13), it suffices to show that, for any xc(g), (15)i=1nXni(j)0a.s.,asn,j=1,2,3,4.

By Eɛn=0 and E|X|p<, we can see that (16)|i=1nEXni(1)|i=1n[n-δP(|Rni|>n-δ)+E|Rni|I(|Rni|>n-δ)]i=1nn-δE|Rni|pn-pδCnδ(p-1)i=1n|Wni(x)|pE|X|pCnδ(p-1)(max1in|Wni(x)|)p-1i=1n|Wni(x)|Cn-(p-1)(s-δ)0,asn. Hence, to prove i=1nXni(1)0a.s., it suffices to show that (17)n=1P(|i=1n(Xni(1)-EXni(1))|>ɛ)<. It is easily seen that, for fixed n1 and xc(g), {Xni(1),1in} are still ND random variables by Lemma 2. Applying Lemma 3, we have for q2 that (18)n=1P(|i=1n(Xni(1)-EXni(1))|>ɛ)Cn=1i=1nE|Xni(1)|q+Cn=1(i=1nE|Xni(1)|2)q/2CI11+CI12. Taking q>max{p,(s+1+p(δ-s))/δ,2/((s-δ)(p-1)+δ),2/s}, we can get that (19)(q-p)δ+s(p-1)>1,qs2>1,[(2-p)δ+s(p-1)]q2=[(s-δ)(p-1)+δ]q2>1.

For I11, we have by Cr's inequality, Lemma 5, and condition (A2) that (20)I11Cn=1i=1n[n-qδP(|Rni|>n-δ)dd.dddd+E|Rni|qI(|Rni|n-δ)]Cn=1i=1n[n-qδP(|Wni(x)X|>n-δ)dddddd+E|Wni(x)X|qI(|Wni(x)X|n-δ)]Cn=1i=1nn-qδE|Wni(x)X|pn-pδCn=1n-(q-p)δ(max1in|Wni(x)|)p-1×i=1n|Wni(x)|Cn=1n-[(q-p)δ+s(p-1)]<.

For I12, if 1<p<2, we have by the proof of (20) that (21)I12Cn=1{i=1n[n-2δP(|Rni|>n-δ)ddddddd+E|Rni|2I(|Rni|n-δ)]i=1n}q/2Cn=1n-[(2-p)δ+s(p-1)]q/2<; if p2, we have by EX2< that (22)I12Cn=1{i=1n[n-2δP(|Rni|>n-δ)ddddddddddd+E|Rni|2I(|Rni|n-δ)]i=1n}q/2Cn=1{i=1n[n-2δP(|Wni(x)X|>n-δ)+Wni2(x)]}q/2Cn=1n-[(2-p)δ+s(p-1)]q/2+Cn=1n-qs/2<. By (18)–(22), we can get (17), which together with (16) imply that i=1nXni(1)0a.s.

Next, we will prove that i=1nXni(2)0a.s. Since 0Xni(2)<ɛ/N, |i=1nXni(2)|=i=1nXni(2)>ɛ implies that there are at least N  i’s such that Xni(2)0. Hence, (23)P(|i=1nXni(2)|>ɛ)P(thereareatleastNi’ssuchthatXni(2)0)1i1<i2<<iNnP(Xni1(2)0,Xni2(2)0,ddddddddddddddd,XniN(2)0)1i1<i2<<iNnP(Rni1>n-δ,Rni2>n-δ,,RniN>n-δ)1i1<i2<<iNnP(Rni1>n-δ)P(Rni2>n-δ)dddddddddddddP(RniN>n-δ)[i=1nP(Rni>n-δ)]N[i=1nP(|Rni|>n-δ)]NC[i=1nE|Wni(x)X|pn-pδ]NC[npδ(max1in|Wni(x)|)p-1i=1n|Wni(x)|]NCn-[(s-δ)p+s]N, which is summable if we choose large integer N such that [(s-δ)p+s]N>1. Hence, i=1nXni(2)0a.s. by Borel-Cantelli lemma.

Since -ɛ/N<Xni(3)0, |i=1nXni(3)|=-i=1nXni(3)>ɛ implies that there are at least Ni’s such that Xni(3)0. Similarly, we have i=1nXni(3)0a.s. as n.

Finally, we will prove that i=1nXni(4)0a.s. as n. It is easily seen that (24)|i=1nXni(4)|i=1n|Rni|I(|Rni|ɛN)+n-δi=1nI(|Rni|ɛN)Cn-si=1n|ɛni|I(|ɛni|Cns)+n-δi=1nI(|ɛni|Cns)Cn-si=1n|ɛni|I(|ɛni|Cis)+n-δi=1nI(|ɛni|Cis)Hn1+Hn2. To prove that i=1nXni(4)0a.s. as n, it suffices to show that Hn10a.s. and Hn20a.s. as n. Firstly, we will show that (25)i=1i-s|ɛni|I(|ɛni|Cis)<a.s.,(26)i=1i-δI(|ɛni|Cis)<a.s. Denote that Tm=i=1mi-s|ɛni|I(|ɛni|Cis). For mk1, it follows by Lemma 5 and E|X|p< that (27)E|Tm-Tk|=i=k+1mi-sE|ɛni|I(|ɛni|Cis)Ci=k+1mi-sE|X|I(|X|Cis)Ci=k+1mi-spCk-sp+10ask, which implies that {Tm,m1} is a Cauchy sequence in L1, and hence there exists a random variable T such that E|T|< and E|Tm-T|0. It follows by Lemma 5 and E|X|p< again that (28)P(|T2k-T|>ɛ)CE|T2k-T|ClimsupmE|T2k-Tm|Ci=2k+1i-sE|ɛni|I(|ɛni|Cis)Ci=2k+1i-sE|X|I(|X|Cis)Ci=2k+1i-spC2-k(sp-1),P(max2k-1<m2k|Tm-T2k-1|>ɛ)CE(max2k-1<m2k|Tm-T2k-1|)Ci=2k-1+12ki-sE|ɛni|I(|ɛni|Cis)Ci=2k-1+12ki-spC2-k(sp-1), which together imply that TmTa.s., and thus (25) holds. Similarly, we can get (26). Therefore, Hn10a.s. and Hn20a.s. follow by (25)-(26) and the Kronecker's lemma immediately. This completes the proof of the theorem.

As an application of Theorem 6, we give the strong consistency for the nearest neighbor estimator of g(x). Without loss of generality, putting A=[0,1], taking xni=i/n, i=1,2,,n, and n1. For any xA, we rewrite |xn1-x|,|xn2-x|,,|xnn-x| as follows: (29)|xR1(x)(n)-x||xR2(x)(n)-x||xRn(x)(n)-x|. If |xni-x|=|xnj-x|, then |xni-x| is permuted before |xnj-x| when xni<xnj.

Let 1knn, the nearest neighbor weight function estimator of g(x) in model (1) is defined as follows: (30)g~n(x)=i=1nW~ni(x)Yni, where (31)W~ni(x)={1kn,if  |xni-x||xRkn(x)(n)-x|,0,otherwise.

Based on the notations above, we can get the following result by using Theorem 6.

Corollary 7.

Let {ɛn,n1} be a sequence of mean zero ND random variables, which is stochastically dominated by a random variable X. Assume that g is continuous on the compact set A. If E|X|p< for some p>1 and if there exists some s(1/p,1) such that kn=[ns], then for any xc(g), (32)g~n(x)g(x)a.s.,asn.

Proof.

It suffices to show that the conditions of Theorem 6 are satisfied. Since g is continuous on the compact set A, hence g is uniformly continuous on the compact set A, which implies that {|g(xni)-g(x)|:1in,n1} is bounded on the set A.

For any x[0,1], if follows from the definition of Ri(x) and W~ni(x) that (33)i=1nW~ni(x)=i=1nW~nRi(x)(x)=i=1kn1kn=1,max1inW~ni(x)=1kn,W~ni(x)0,i=1n|W~ni(x)|·|g(xni)-g(x)|I(|xni-x|>a)Ci=1n(xni-x)2|W~ni(x)|a2=Ci=1kn(xRi(x)(n)-x)2kna2Ci=1kn(i/n)2kna2C(knna)2,a>0. Hence, conditions (A1)–(A3) and (8) are satisfied. By Theorem 6, we can get (32) immediately. This completes the proof of the corollary.

Acknowledgments

The authors are most grateful to the Editor Michiel Bertsch and anonymous referee for careful reading of the paper 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 (1208085QA03, 1308085QA03), Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), Doctoral Research Start-up Funds Projects of Anhui University, Students Innovative Training Project of Anhui University (201310357004), and the Students Science Research Training Program of Anhui University (KYXL2012007, kyxl2013003).

Georgiev A. A. Grossmann W. Local properties of function fitting estimates with applications to system identification Mathematical Statistics and Applications 1985 Dordrecht, The Netherlands Reidel 141 151 Volume 2 of Proceedings of the 4th Pannonian Symposium on Mathematical Statistics, Bad Tatzmannsdorf, Austria, 4–10 September 1983 Georgiev A. A. Greblicki W. Nonparametric function recovering from noisy observations Journal of Statistical Planning and Inference 1986 13 1 1 14 10.1016/0378-3758(86)90114-X MR822121 ZBL0596.62041 Georgiev A. A. Consistent nonparametric multiple regression: the fixed design case Journal of Multivariate Analysis 1988 25 1 100 110 10.1016/0047-259X(88)90155-8 MR935297 ZBL0637.62044 Müller H. G. Weak and universal consistency of moving weighted averages Periodica Mathematica Hungarica 1987 18 3 241 250 10.1007/BF01848087 MR902526 ZBL0596.62040 Fan Y. Consistent nonparametric multiple regression for dependent heterogeneous processes: the fixed design case Journal of Multivariate Analysis 1990 33 1 72 88 10.1016/0047-259X(90)90006-4 MR1057055 ZBL0698.62040 Roussas G. G. Consistent regression estimation with fixed design points under dependence conditions Statistics & Probability Letters 1989 8 1 41 50 10.1016/0167-7152(89)90081-3 MR1006420 ZBL0674.62026 Roussas G. G. Tran L. T. Ioannides D. A. Fixed design regression for time series: asymptotic normality Journal of Multivariate Analysis 1992 40 2 262 291 10.1016/0047-259X(92)90026-C MR1150613 ZBL0764.62073 Tran L. Roussas G. Yakowitz S. Truong Van B. Fixed-design regression for linear time series The Annals of Statistics 1996 24 3 975 991 10.1214/aos/1032526952 MR1401833 ZBL0862.62069 Hu S. H. Zhu C. H. Chen Y. B. Wang L. C. Fixed-design regression for linear time series Acta Mathematica Scientia B 2002 22 1 9 18 MR1883874 ZBL1010.62085 Hu S. H. Pan G. M. Gao Q. B. Estimation problems for a regression model with linear process errors Applied Mathematics-A Journal of Chinese Universities 2003 18 1 81 90 MR1960506 Liang H.-Y. Jing B.-Y. Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences Journal of Multivariate Analysis 2005 95 2 227 245 10.1016/j.jmva.2004.06.004 MR2170396 ZBL1070.62022 Yang W. Z. Wang X. J. Wang X. H. Hu S. H. The consistency for estimator of nonparametric regression model based on NOD errors Journal of Inequalities and Applications 2012 2012 article 140 10.1186/1029-242X-2012-140 Shen A. T. Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models Abstract and Applied Analysis 2013 2013 9 862602 10.1155/2013/862602 Lehmann E. L. Some concepts of dependence The Annals of Mathematical Statistics 1966 37 5 1137 1153 MR0202228 10.1214/aoms/1177699260 ZBL0146.40601 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 ZBL0508.62041 Wu Q. Y. Complete convergence for weighted sums of sequences of negatively dependent random variables Journal of Probability and Statistics 2011 2011 16 10.1155/2011/202015 202015 MR2774947 ZBL1221.60041 Volodin A. On the Kolmogorov exponential inequality for negatively dependent random variables Pakistan Journal of Statistics 2002 18 2 249 253 MR1944611 ZBL1128.60304 Asadian N. Fakoor V. Bozorgnia A. Rosenthal's type inequalities for negatively orthant dependent random variables Journal of the Iranian Statistical Society 2006 5 1-2 66 75 Kim H. C. The Hájeck-Rènyi inequality for weighted sums of negatively orthant dependent random variables International Journal of Contemporary Mathematical Sciences 2006 1 5–8 297 303 MR2289035 ZBL1156.60306 Amini M. Azarnoosh H. A. Bozorgnia A. The strong law of large numbers for negatively dependent generalized Gaussian random variables Stochastic Analysis and Applications 2004 22 4 893 901 10.1081/SAP-120037623 MR2062950 ZBL1056.60024 Amini M. Zarei H. Bozorgnia A. Some strong limit theorems of weighted sums for negatively dependent generalized Gaussian random variables Statistics & Probability Letters 2007 77 11 1106 1110 10.1016/j.spl.2007.01.015 MR2395067 ZBL1120.60022 Ko M. H. Kim T. S. Almost sure convergence for weighted sums of negatively orthant dependent random variables Journal of the Korean Mathematical Society 2005 42 5 949 957 10.4134/JKMS.2005.42.5.949 MR2157354 Klesov O. Rosalsky A. Volodin A. I. On the almost sure growth rate of sums of lower negatively dependent nonnegative random variables Statistics & Probability Letters 2005 71 2 193 202 10.1016/j.spl.2004.10.027 MR2126775 ZBL1070.60030 Wang X. J. Hu S. H. Shen A. T. Yang W. Z. An exponential inequality for a NOD sequence and a strong law of large numbers Applied Mathematics Letters 2011 24 2 219 223 10.1016/j.aml.2010.09.007 MR2735145 ZBL1205.60068 Amini M. Bozorgnia A. Complete convergence for negatively dependent random variables Journal of Applied Mathematics and Stochastic Analysis 2003 16 2 121 126 10.1155/S104895330300008X MR1989578 Kuczmaszewska A. On some conditions for complete convergence for arrays of rowwise negatively dependent random variables Stochastic Analysis and Applications 2006 24 6 1083 1095 10.1080/07362990600958754 MR2273771 ZBL1108.60021 Taylor R. L. Patterson R. F. Bozorgnia A. A strong law of large numbers for arrays of rowwise negatively dependent random variables Stochastic Analysis and Applications 2002 20 3 643 656 10.1081/SAP-120004118 MR1900307 ZBL1003.60032 Zarei H. Jabbari H. Complete convergence of weighted sums under negative dependence Statistical Papers 2011 52 2 413 418 10.1007/s00362-009-0238-4 MR2795888 ZBL1247.60044 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 ZBL1202.60050 Sung S. H. Complete convergence for weighted sums of negatively dependent random variables Statistical Papers 2012 53 1 73 82 10.1007/s00362-010-0309-6 MR2878592 ZBL06054163 Wang X. J. Hu S. H. Yang W. Z. Complete convergence for arrays of rowwise negatively orthant dependent random variables Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A 2012 106 2 235 245 10.1007/s13398-011-0048-0 MR2978912 ZBL1260.60062 Wang X. J. Hu S. H. Yang W. Z. Ling N. X. Exponential inequalities and inverse moment for NOD sequence Statistics & Probability Letters 2010 80 5-6 452 461 10.1016/j.spl.2009.11.023 MR2593586 ZBL1186.60015 Shen A. T. Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables Journal of Inequalities and Applications 2011 2011 article 93 10.1186/1029-242X-2011-93 MR2853335 ZBL06198001 Shen A. T. On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales A 2013 107 2 257 271 10.1007/s13398-012-0067-5 Bozorgnia A. Patterson R. F. Taylor R. L. Limit theorems for dependent random variables 1 Proceeding of the World Congress Nonlinear Analysts (WCNA '92) 1992 Berlin, Germany de Gruyter 1639 1650 MR1389197 ZBL0845.60010 Wu Q. Y. A strong limit theorem for weighted sums of sequences of negatively dependent random variables Journal of Inequalities and Applications 2010 2010 8 383805 10.1155/2010/383805 MR2678912 ZBL1202.60044 Wu Q. Y. A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables Journal of Inequalities and Applications 2012 2012 article 50 10.1186/1029-242X-2012-50 MR2928247 ZBL06210053 Shen A. T. Wu R. C. Strong and weak convergence for asymptotically almost negatively associated random variables Discrete Dynamics in Nature and Society 2013 2013 7 235012 MR3037709 ZBL1269.60036 10.1155/2013/235012