Review Article Complete Convergence for Negatively Dependent Sequences of Random Variables

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1 Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 010, Article ID 50793, 10 pages doi: /010/50793 Review Article Complete Covergece for Negatively Depedet Sequeces of Radom Variables Quyig Wu College of Sciece, Guili Uiversity of Techology, Guili , Chia Correspodece should be addressed to Quyig Wu, Received 5 Jauary 010; Accepted 6 March 010 Academic Editor: Jewgei Dshalalow Copyright q 010 Quyig Wu. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We study the complete covergece for egatively depedet sequeces of radom variables. As a result, we exted some complete covergece theorems for idepedet radom variables to the case of egatively depedet radom variables without ecessarily imposig ay extra coditios. 1. Itroductio ad Lemmas Defiitio 1.1. Radom variables X ad Y are said to egatively depedet ND if P X x, Y y PX xp Y y 1.1 for all x, y R. A collectio of radom variables is said to be pairwise egatively depedet PND if every pair of radom variables i the collectio satisfies 1.1. It is importat to ote that 1.1 implies P X>x,Y>y PX >xp Y>y 1. for all x, y R. Moreover, it follows that 1. implies 1.1, ad hece, 1.1 ad 1. are equivalet. Ebrahimi ad Ghosh 1 showed that 1.1 ad 1. are ot equivalet for a collectio of 3 or more radom variables. They cosidered radom variables X 1,X,ad X 3 where X 1,X,X 3 assumed the values 0, 1, 1, 1, 0, 1, 1, 1, 0, ad0, 0, 0 each with probability 1/4. The radom variables X 1,X,adX 3 are pairwise idepedet, ad hece, they satisfy both 1.1 ad 1. for all pairs. However, PX 1 >x 1,X >x,x 3 >x 3 PX 1 >x 1 PX >x PX 3 >x 3 1.3

2 Joural of Iequalities ad Applicatios for all x 1, x,adx 3,but PX 1 0,X 0,X /1 8 PX 1 0PX 0PX Placig probability 1/4 o each of the other vertices {1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1} provides the coverse example of pairwise idepedet radom variables which will ot satisfy 1.3 with x 1 0, x 0, ad x 3 0 but where the desired ipx 1 x 1,X x,x 3 x 3 3 i1 PX i x i hold for all x 1,x,adx 3. Cosequetly, the followig defiitio is eeded to defie sequeces of egatively depedet radom variables. Defiitio 1.. Radom variables X 1,...,X are said to be egatively depedet ND if for all real x 1,...,x, P Xj x j P Xj >x j P [ ] X j x j, P X j >x j. 1.5 A ifiite sequece of radom variables {X ; 1} is said to be ND if every fiite subset X 1,...,X is ND. Defiitio 1.3. Radom variables X 1,X,...,X, are said to be egatively associated NA if for every pair of disjoit subsets A 1 ad A of {1,,...,}, cov f 1 X i ; i A 1,f Xj ; j A 0, 1.6 where f 1 ad f are icreasig for every variable or decreasig for every variable, such that this covariace exists. A ifiite sequece of radom variables {X ; 1} is said to be NA if every fiite subfamily is NA. The defiitio of PND is give by Lehma. The cocept of ND is give by Bozorgia et al. 3, ad the defiitio of NA is itroduced by Joag-Dev ad Proscha 4. These coceptios of depedece radom variables have bee very useful i reliability theory ad applicatios. It is easy to see that NA implies ND from the defiitios. But i the followig example, we will show that ND does ot imply NA. Example 1.4. Let X i be a biary radom variable such that PX i 0 PX i for i 1,, 3. Let X 1,X,X 3 take the values 0, 0, 1, 0, 1, 0, 1, 0, 0, ad1, 1, 1, each with probability 1/4. It ca be verified that all the ND coditios hold. However, PX 1 X 3 1,X /3 8 PX 1 X 3 1PX Thus, X 1,X,X 3 are ot NA.

3 Joural of Iequalities ad Applicatios 3 From the above example, it is show that ND is much weaker tha NA. Because of the wide applicatios of ND radom variables, the otios of ND radom variables have received more ad more attetio recetly. A series of useful results have bee established cf. 3, Hece, it is highly desirable ad of cosiderable sigificace to exted the limit properties of idepedet or NA radom variables to the case of ND radom variables theorems ad applicatios. Complete covergece is oe of the most importat problems i probability theory. Recet results of the complete covergece ca be foud i Wu 11, 1 ad Sug 13, 14.I this paper we study the complete covergece for egatively depedet radom variables. As a result, we exted some complete covergece theorems for idepedet radom variables to the egatively depedet radom variables without ecessarily imposig ay extra coditios. Lemma 1.5 see 3. Let X 1,...,X be ND radom variables ad let {f ; 1} be a sequece of Borel fuctios all of which are mootoe icreasig or all are mootoe decreasig. The {f X ; 1} is a sequece of ND r.v. s. Lemma 1.6 see 3. Let X 1,...,X be oegative r.v. s which are ND. The E X j EX j. 1.8 I particular, let X 1,...,X be ND ad let t 1,...,t be all oegative or opositive real umbers. The E exp t j X j E exp t j X j. 1.9 Lemma 1.7. Let {X ; 1} be a sequece of ND radom variables with EX i 0, EX i <.The for 1, ES EX k, 1.10 where S X k. Proof. Obviously, ND implies PND from their defiitios. Thus, by Lemma of Wu 1, Lemma 1.7 holds.. Mai Results ad the Proof I the followig, let a b deote that there exists a costat c>0 such that a cb for sufficietly large,logx mea lmaxx, e,ads X j.

4 4 Joural of Iequalities ad Applicatios Theorem.1. Let {X ; 1} be a sequece of ND idetically distributed radom variables. Let for some 0 <α 1 E X 1 /α <, EX 1 0,.1 a c α for k ad some 0 <c<, a 0 for k>,. c a log o 1..3 The c T a X k 0,.4 where c deotes complete covergece. Theorem.. Let {X ; 1} be a sequece of ND idetically distributed radom variables with E X 1 /α <, for some α>1..5 The S α c 0..6 Remark.3. Theorems.1 ad. exted correspodig results for idepedet r.v.s. to ND r.v.s. without ecessarily addig ay extra coditios. Proof of Theorem.1. By /α ad.1, we have EX 1 <. Letε > 0 be give. By T a X k a X k, where a maxa, 0 0, ad a max a, 0 0, without loss of geerality, we ca assume that a > 0 for all 1, k, ad EX 1 1. For k, let X 1 α/3 a 1 I X k < α/3 a 1 X k I Xk α/3 a 1 α/3 a 1 I X k > α/3 a 1, X X k α/3 a 1 I Xk < εa 1 /4 X k α/3 a 1 I Xk >εa 1 /4, X 3 X k X 1 k X k X k α/3 a 1 I εa 1 /4<X k< α/3 a 1 T i X k α/3 a 1 a X i, i 1,, 3. I α/3 a 1 <X k<εa 1 /4,.7

5 Joural of Iequalities ad Applicatios 5 Thus, T T 1 T T 3..8 Note that T 1 T T 3 T > 3ε >ε >ε >ε..9 We shall prove that 1 P T > 3ε < by provig that T i P >ε <, i 1,, By EX k 0ad., a EX 1 a E X k X 1 ET 1 a E X k α/3 a 1 I Xk < α/3 a 1 E X k α/3 a 1 I Xk > α/3 a 1 a E X 1 I X1 > α/3 a 1.11 So to prove 1 Let X 1 α1 E X 1 I X1 >c 1 α/3 1α/3 E X 1 /α 0,. 1 P T >ε < it suffices to show that T 1 P 1 X1 EX1, T 1 T 1 ua X 1 1, E X 1 0, ad E X 1 EX 1 E exp ua X 1 E 1 1 E ET 1 > ε <..1 ET 1.Fix 1. Let u miε/4c, α/3 /. Sice 1, it follows that ua X 1 j! ua X 1 j E ua X 1 u a E X e.5 1 u a u exp a. 1 1 j! j3.13

6 6 Joural of Iequalities ad Applicatios Sice X 1 ad X1 are odecreasig ad oicreasig fuctios of X k, respectively, thus {ua X 1, 1, k } ad {ua X 1, 1, k } are also ND by Lemma 1.5. It follows from Lemma 1.6 ad.13 that By the Markov iequality, E exp u T 1 E exp ua X 1 E exp ua X 1 P T 1 >ε exp εue exp exp u a exp u c. ut 1.14 exp εu u c..15 Sice { X 1 } is also satisfyig the coditios: ua X 1 1, E X 1 0, ad E X 1, replacig the X k by X k i.15 the argumet will the establish P T 1 < ε exp εu u c..16 Thus, P T 1 > ε exp εu u c..17 If ε/c > α/3, the by the defiitio of u, we have u α/3 /, εu/ u c ε α/3 /8. Hece T 1 P > ε exp εα/3 < If ε/c α/3, the by the defiitio of u, we have u ε/4c, ad εu/ u c ε /16c. Ad c <ε /3 log for sufficietly large from c olog 1. Hece T 1 P > ε exp ε 1 16c <. 1 Now we prove that 1 T 1 P T >ε <. Sice a X X a 1 a X k I Xk >εa 1 /4,.19.0

7 Joural of Iequalities ad Applicatios 7 therefore, T >ε X k > εa 1 4 X k >ε α 4c 1..1 Sice 0 <a c α for k, thus T P >ε P X k >ε α 4c 1 P X 1 >ε α 4c 1 B α,. where B ε 1/α 4c 1/α > 0. Therefore T P >ε P X 1 1/α >B 1 1 P Bj < X 1 1/α j 1 B 1 j 1 j P Bj < X 1 1/α j 1 B j P Bj < X 1 1/α j 1 B B E X 1 /α I Bj< X1 1/α B.3 E X 1 /α <. Lastly, we prove that 1 T 3 >ε 3 P T >ε <. Sice there exist at least 4 idices k such that X k >a 1 α/3,.4 we have T 3 P >ε P there exist at least 4 idices k such that a X k > α/3 1i 1 < <i 4 P a i1 X i1 > α/3,a i X i > α/3,a i3 X i3 > α/3,a i4 X i4 > α/3..5

8 8 Joural of Iequalities ad Applicatios By the defiitio of ND, ad the fact that 0 <a c α for k, we coclude that T 3 P >ε 4 Xij P >c 1 α/3 1i 1 < <i 4.6 C 4 P 4 X 1 >c 1 α/3 4 P 4 X 1 >c 1 α/3. Thus, by.1 T 3 P >ε 4 α/3/α E X 1 /α E X 1 /α 4 4/3 <. 1.7 Together with.19.7, weget P T >ε <, 1.8 for all ε>0 as desired. This completes the proof of Theorem.1. Proof of Theorem.. Let X k α I Xk < α X k I Xk α α I Xk > α, 1, k, S k X k..9 If /α 1, the E X 1 < ad P X 1 > α 0from.5. Thus α ES k α E Xk I Xk α α P X k > α E X 1 1 α P X 1 > α If /α < 1, from above proof, we have α ES k 1 α E X 1 I k 1 α < X 1 k α P X 1 > α..31

9 Joural of Iequalities ad Applicatios 9 Sice k 1 α E X 1 I k 1 α < X 1 k α k 1 α E X 1 /α k α1 /α I k 1< X1 1/α k k 1 E X 1 /α I k 1< X1 1/α k.3 E X 1 /α <, by α>1, ad the Kroecker lemma, combiig with.31, weget α ES k 0. Thus, for sufficietly large, α ES k < ε..33 Sice X is odecreasig fuctio of X k,thus{x, 1, k } is also ND by Lemma 1.5. It follows from Lemma 1.7,.13, /α <, the Markov iequality, ad 1 P X 1 > α E X 1 /α <, that P S >ε α P X k > α P S k ES k >ε α ES k P X 1 > α P S k ES k > εα E X 1 /α 1 ES α k ES k α 1 α1 EX 1 I X 1 < α α P X 1 > α 1 1 EX k α1 EX 1 I k 1 α X 1 <k α P X 1 > α k 1 α1 E X 1 /α k α /α I k 1 α X 1 <k α E X 1 /α k α E X 1 /α k α /α I k 1 α X 1 <k α E X 1 /α I k 1 α X 1 <k α.34 <.

10 10 Joural of Iequalities ad Applicatios Hece, S α c This completes the proof of Theorem.. Ackowledgmets The author is grateful to the referees ad the editors for their valuable commets ad some helpful suggestios that improved the clarity ad readability of the paper. This work was supported by the Natioal Natural Sciece Foudatio of Chia , the Support Program of the New Cetury Guagxi Chia Te-hudred-thousad Talets Project 00514, ad the Guagxi Chia Sciece Foudatio Refereces 1 N. Ebrahimi ad M. Ghosh, Multivariate egative depedece, Commuicatios i Statistics A, vol. 10, o. 4, pp , E. L. Lehma, Some cocepts of depedece, Aals of Mathematical Statistics, vol.37,o.5,pp , A. Bozorgia, R. F. Patterso, ad R. L. Taylor, Limit theorems for ND r.v. s, Tech. Rep., Uiversity of Georgia, Athes, Greece, K. Joag-Dev ad F. Proscha, Negative associatio of radom variables, with applicatios, The Aals of Statistics, vol. 11, o. 1, pp , A. Bozorgia, R. F. Patterso, ad R. L. Taylor, Weak laws of large umbers for egatively depedet radom variables i Baach spaces, i Research Developmets i Probability ad Statistics, E. Bruer ad M. Deer, Eds., pp. 11, VSP Iteratioal Sciece, Vilius, Lithuaia, M. Amii, Some cotributio to limit theorems for egatively depedet radom variable, Ph.D. thesis, V. Fakoor ad H. A. Azaroosh, Probability iequalities for sums of egatively depedet radom variables, Pakista Joural of Statistics, vol. 1, o. 3, pp , H. R. Nili Sai, M. Amii, ad A. Bozorgia, Strog laws for weighted sums of egative depedet radom variables, Joural of Scieces, vol. 16, o. 3, pp , O. Klesov, A. Rosalsky, ad A. I. Volodi, O the almost sure growth rate of sums of lower egatively depedet oegative radom variables, Statistics & Probability Letters, vol. 71, o., pp , Q. Y. Wu ad Y. Y. Jiag, Strog cosistecy of M estimator i liear model for egatively depedet radom samples, Commuicatios i Statistics Theory ad Methods. I press. 11 Q. Y. Wu, Complete covergece for weighted sums of sequeces of egatively depedet radom variables, Joural of Probability ad Statistics. I press. 1 Q. Y. Wu, Covergece properties of pairwise NQD radom sequeces, Acta Mathematica Siica, vol. 45, o. 3, pp , S. H. Sug, Maximal iequalities for depedet radom variables ad applicatios, Joural of Iequalities ad Applicatios, vol. 008, Article ID , 10 pages, S. H. Sug, Momet iequalities ad complete momet covergece, Joural of Iequalities ad Applicatios, vol. 009, Article ID 7165, 14 pages, 009.