Research Article Strong and Weak Convergence for Asymptotically Almost Negatively Associated Random Variables

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1 Discrete Dyamics i Nature ad Society Volume 2013, Article ID , 7 pages Research Article Strog ad Weak Covergece for Asymptotically Almost Negatively Associated Radom Variables Aitig She ad Rachao Wu School of Mathematical Sciece, Ahui Uiversity, Hefei , Chia Correspodece should be addressed to Aitig She; empress201010@126.com Received 10 December 2012; Accepted 16 Jauary 2013 Academic Editor: Bigge Zhag Copyright 2013 A. She ad R. 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. The strog law of large umbers for sequeces of asymptotically almost egatively associated (AANA, i short) radom variables is obtaied, which geeralizes ad improves the correspodig oe of Bai ad Cheg (2000) for idepedet ad idetically distributed radom variables to the case of AANA radom variables. I additio, the Feller-type weak law of large umber for sequeces of AANA radom variables is obtaied, which geeralizes the correspodig oe of Feller (1946) for idepedet ad idetically distributed radom variables. 1. Itroductio May useful liear statistics based o a radom sample are weighted sums of idepedet ad idetically distributed radom variables. Examples iclude least-squares estimators, oparametric regressio fuctio estimators, ad ackkife estimates,. I this respect, studies of strog laws for these weighted sums have demostrated sigificat progress i probability theory with applicatios i mathematical statistics. Let {X, 1}be a sequece of radom variables ad let {a i,1 i, 1}be a array of costats. A commo expressio for these liear statistics is T = a ix i.some recet results o the strog law for liear statistics T ca be foud i Cuzick [1],Baietal.[2], Bai ad Cheg [3], Cai [4], Wu [5], Sug [6], Zhou et al. [7], ad Wag et al. [8]. Our emphasis i this paper is focused o the result of Bai ad Cheg [3].Theygavethefollowigtheorem. Theorem A. Suppose that 1 < α,β <, 1 p < 2, ad 1/p = 1/α + 1/β. Let{X, X, 1} be a sequece of idepedet ad idetically distributed radom variables satisfyig EX = 0,adlet{a k,1 k, 1}be a array of real costats such that lim sup( 1 a k α ) 1/α <. (1) If E X β <,the lim 1/p a k X k =0 a.s. (2) We poit out that the idepedece assumptio is ot plausible i may statistical applicatios. So it is of iterest to exted the cocept of idepedece to the case of depedece. Oe of these depedece structures is asymptotically almost egatively associated, which was itroduced by Chadra ad Ghosal [9] as follows. Defiitio 1. Asequece{X, 1}of radom variables is called asymptotically almost egatively associated (AANA, i short) if there exists a oegative sequece u() 0 as such that Cov (f (X ),g(x +1,X +2,...,X +k )) u() [Var (f (X )) Var (g (X +1,X +2,...,X +k ))] 1/2, (3) for all, k 1 ad for all coordiatewise odecreasig cotiuous fuctios f ad g wheever the variaces exist. It is easily see that the family of AANA sequece cotais egatively associated (NA, i short) sequeces (with u() = 0, 1) ad some more sequeces of radom

2 2 Discrete Dyamics i Nature ad Society variables which are ot much deviated from beig egatively associated. A example of a AANA sequece which is ot NA was costructed by Chadra ad Ghosal [9]. Hece, extedig the limit properties of idepedet or NA radom variables to the case of AANA radom variables is highly desirable i the theory ad applicatio. Sice the cocept of AANA sequece was itroduced by Chadra ad Ghosal [9], may applicatios have bee foud. See, for example, Chadra ad Ghosal [9] derived the Kolmogorov type iequality ad the strog law of large umbers of Marcikiewicz-Zygmud; Chadra ad Ghosal [10] obtaied the almost sure covergece of weighted averages; Wag et al. [11] established the law of the iterated logarithm for product sums; Ko et al. [12]studied the Háek- Réyi type iequality; Yua ad A [13] establishedsome Rosethal type iequalities for maximum partial sums of AANA sequece; Wag et al. [14] obtaied some strog growth rate ad the itegrability of supremum for the partial sums of AANA radom variables; Wag et al. [15, 16]studied complete covergece for arrays of rowwise AANA radom variables ad weighted sums of arrays of rowwise AANA radom variables, respectively; Hu et al. [17] studiedthe strog covergece properties for AANA sequece; Yag et al. [18] ivestigated the complete covergece, complete momet covergece, ad the existece of the momet of supermum of ormed partial sums for the movig average process for AANA sequece, ad so forth. The mai purpose of this paper is to study the strog covergece for AANA radom variables, which geeralizes ad improves the result of Theorem A. I additio, we will give the Feller-type weak law of large umber for sequeces of AANA radom variables, which geeralizes the correspodig oe of Feller [19] for idepedet ad idetically distributed radom variables. Throughout this paper, let {X, 1} be a sequece of AANA radom variables with the mixig coefficiets {u(), 1}. S = X i.fors > 1,lett s/(s 1) be the dual umber of s. The symbol C deotes a positive costat which may be differet i various places. Let I(A) be the idicator fuctio of the set A. a =O(b ) stads for a Cb. The defiitio of stochastic domiatio will be used i the paper as follows. Defiitio 2. Asequece{X, 1}of radom variables is said to be stochastically domiated by a radom variable X if there exists a positive costat C such that for all x 0ad 1. P( X >x) CP( X >x), (4) Our mai results are as follows. Theorem 3. Suppose that 0 < α,β <, 0 < p < 2, ad 1/p = 1/α + 1/β. Let{X, 1} be a sequece of AANA radom variables, which is stochastically domiated by a radom variable X ad EX =0,ifβ>1. Suppose that there exists a positive iteger k such that =1 u1/(1 s) () < for some s (3 2 k 1,4 2 k 1 ] ad s>1/(mi{1/2, 1/α, 1/β, 1/p 1/2}). Let{a i,i 1, 1} be a array of real costats satisfyig If E X β <,the lim 1/p max 1 a i α =O(). (5) a i X i = 0 a.s. (6) Remark 4. Theorem 3 geeralizes ad improves Theorem A of Bai ad Cheg [3] for idepedet ad idetically distributed radom variables to the case of AANA radom variables, sice Theorem 3 removes the idetically distributed coditio ad expads the rages α, β, ad p, respectively. At last, we will preset the Feller-type weak law of large umber for sequeces of AANA radom variables, which geeralizes the correspodig oe of Feller [19] for idepedet ad idetically distributed radom variables. Theorem 5. Let α>1/2ad {X, X, 1} be a sequece of idetically distributed AANA radom variables with the mixig coefficiets {u(), 1} satisfyig =1 u2 () <. If the 2. Preparatios lim P ( X >α )=0, (7) S α 1 α EXI ( X α ) P 0. (8) To prove the mai results of the paper, we eed the followig lemmas. The first two lemmas were provided by Yua ad A [13]. Lemma 6 (cf. see [13, Lemma2.1]).Let {X, 1} be a sequece of AANA radom variables with mixig coefficiets {u(), 1}, f 1,f 2,... be all odecreasig (or all oicreasig) cotiuous fuctios, the {f (X ), 1}is still a sequece of AANA radom variables with mixig coefficiets {u(), 1}. Lemma 7 (cf. see [13,Theorem2.1]). Let p>1ad {X, 1} be a sequece of zero mea radom variables with mixig coefficiets {u(), 1}. If =1 u2 () <, the there exists a positive costat C p depedig oly o p such that for all 1ad 1<p 2, E(max 1 p X i ) C p E X i p. (9)

3 Discrete Dyamics i Nature ad Society 3 If =1 u1/(p 1) () < for some p (3 2 k 1,4 2 k 1 ], where iteger umber k 1, the there exists a positive costat D p depedig oly o p such that for all 1, E(max 1 p { X i ) D p { E X i p +( { p/2 } EX 2 i ) }. } (10) The last oe is a fudametal property for stochastic domiatio. The proof is stadard, so the details are omitted. Lemma 8. Let {X, 1}be a sequece of radom variables, which is stochastically domiated by a radom variable X. The for ay α>0ad b>0, E X α I( X b) C 1 [E X α I ( X b) +b α P ( X >b)], E X α I( X >b) C 2E X α I ( X >b), where C 1 ad C 2 are positive costats. 3. Proofs of the Mai Results (11) Proof of Theorem 3. Without loss of geerality, we assume that a i 0(otherwise, we use a + i ad a i istead of a i,ad ote that a i =a + i a i ). Deote for 1 i ad 1that Y i = 1/β I(X i < 1/β ) +X i I( X i 1/β )+ 1/β I(X i > 1/β ), Z i =(X i + 1/β )I(X i < 1/β )+(X i 1/β )I(X i > 1/β ). (12) Hece, X i =Y i +Z i, which implies that 1/p max a i X i 1 1/p max 1 a i Z i + 1/p max a i Y i 1 1/p max a i Z i 1 + 1/p max a i EY i 1 + 1/p max 1 H+I+J. a i (Y i EY i ) (13) To prove (6), it suffices to show that H 0a.s., I 0ad J 0a.s. as. Firstly, we will show that H 0a.s. For ay 0 < γ α, it follows from (5) adhölder s iequality that a i γ ( a i α ) γ/α ( 1) 1 γ/α for ay 0<α γ, it follows from (5)agaithat a i γ ( a i α ) Combiig (14) ad(15), we have The coditio E X β <yields that =1 P(Z =0)= γ/α C, (14) C γ/α. (15) a i γ C max(1,γ/α). (16) =1 P( X >1/β ) CP( X > 1/β ) CE X β <, =1 (17) which implies that P(Z =0,i.o.) = 0 by Borel-Catelli lemma. Thus, we have by (5)that H 1/p max a i Z i 1 1/p a iz i C 1/p (max 1 i a i α ) 1/α Z i C 1/p ( C 1/β Secodly, we will prove that a i )1/α α Z i Z i 0 a.s., as. (18) I 1/p max a i EY i 0, as. (19) 1

4 4 Discrete Dyamics i Nature ad Society If 0<β 1,thewehavebyLemma 8 ad (16)that I 1/p 1/p C 1/p C 1/p a iey i =C 1/α 1 E X β a i [E X i I( X i 1/β ) + 1/β P( X i >1/β )] a i [E X I( X 1/β ) + 1/β P( X > 1/β )] a i [(1 β)/β E X β I( X 1/β ) + 1/β 1 E X β I( X > 1/β )] a i C 1/α 1+max(1,1/α) 0, as. If β>1,thewehavebyex =0, Lemma 8 ad (16)that I 1/p C 1/p a iey i a i [E X i I( X i >1/β )+ 1/β P( X i >1/β )] C 1/p C 1/p a i E X I( X >1/β ) a i 1/β 1 E X β I( X > 1/β ) C 1/α 1+max(1,1/α) 0, as. Hece, (19)followsfrom(20)ad(21)immediately. To prove (6), it suffices to show that H 1/p max 1 a i (Y i EY i ) 0 a.s., (20) (21) as. (22) By Borel-Catelli Lemma, we oly eed to show that for ay ε>0, =1 P (max 1 a i (Y i EY i ) >ε 1/p ) <. (23) For fixed 1,itiseasilyseethat{a i (Y i EY i ), 1 i }are still AANA radom variables by Lemma 6.Takig s>1/mi{1/2, 1/α, 1/β, 1/p 1/2} > 2,wehavebyMarkov s iequality ad Lemma 7 that P(max 1 =1 C =1 C a i (Y i EY i ) >ε 1/p ) s/p E(max 1 s/p =1 +C s/p ( =1 J 1 +J 2. a i (Y i EY i ) E a i (Y i EY i ) s E a i (Y i EY i ) 2 ) s ) s/2 (24) For J 1,wehavebyC r iequality, Jese s iequality, (15), ad Lemma 8 that J 1 C C s/p =1 s/p =1 a i s E Y i s a i s [E X i s I( X i 1/β )+ s/β P( X i >1/β )] C s/p =1 a i s [E X s I( X 1/β )+ s/β P( X > 1/β )] C s/β E X s I( X 1/β )+CP( X > 1/β ) =1 C C s/β =1 E X s I =1 ((i 1) 1/β < X i 1/β )+CE X β C =i E X s I((i 1) 1/β < X i 1/β ) s/β +CE X β i (s β)/β E X β I ((i 1) 1/β < X i 1/β )i s/β+1 +CE X β CE X β <. (25)

5 Discrete Dyamics i Nature ad Society 5 Next, we will prove that J 2 <.ByC r iequality, Jese s iequality ad Lemma 8 agai, we ca see that E a i (Y i EY i ) 2 Proof of Theorem 5. Deote for 1 i ad 1that Y i = α I(X i < α )+X i I ( X i α )+ α I(X i > α ) (31) C C a 2 i EY2 i C max(1,2/α) a 2 i [EX2 i I( X i 1/β )+ 2/β P( X i >1/β )] a 2 i [EX2 I( X 1/β )+ 2/β P( X > 1/β )] [EX 2 I( X 1/β )+ 2/β P( X > 1/β )]. (26) ItfollowsbyMarkov siequalityadthefacte X β <that ad T = Y i. By the assumptio (7), we have for ay ε>0that S P( α T α >ε) P(S =T ) P( (X i =Y i )) P( X i >α ) =P( X > α ) 0,, (32) EX 2 I( X 1/β )+ 2/β P( X > 1/β ) (2 β)/β E X β I( X 1/β ) { + 1+2/β E X β ( X > 1/β ), β<2 { { EX 2 I( X 1/β )+EX 2, β 2, (27) which implies that S α T α P 0. (33) { C 1+2/β E X β, β < 2, CEX 2, β 2. Hece, i order to prove (8), we oly eed to show that If we deote δ=max{ 1 + 2/p, 2/β, 2/α, 1},thewecaget by (26)ad(27)that T α ET α P 0. (34) It is easily see that E a i (Y i EY i ) 2 C δ. (28) By (7) agai ad Toeplitz s lemma, we ca get that k2α 2 kp( X >k α ) k2α 2 0,. (35) ( 1 p + δ 2 )s= max { 1 2, 1 α, 1 β, 1 p }s = mi { 1 2, 1 α, 1 β, 1 p 1 2 }s< 1. (29) Hece, we have by (28)ad(29)that Note that k 2α 2 2α 1, for α> 1 2. (36) J 2 C ( 1/p+δ/2)s <, (30) =1 which together with J 1 <yields (23). This completes the proof of the theorem. Combig (35)ad(36), we have 2α+1 k 2α 1 P( X >k α ) 0,. (37)

6 6 Discrete Dyamics i Nature ad Society By Lemma 7 (takig p=2), (7), ad (37), we ca get that P( T ET >εα ) C 2α E T ET 2 C 2α EY 2 i C 2α+1 [EX 2 I( X α )+ 2α P( X > α )] =C 2α+1 EX 2 I ( X α )+CP( X > α ) =C 2α+1 EX 2 I ((k 1) α X k α )+CP( X > α ) C 2α+1 k 2α [P( X > (k 1) α ) P( X >k α )] +CP( X > α ) =C 2α+1 1 [ +CP( X > α ) C 2α+1 [ ((k+1) 2α k 2α )P( X >k α ) +P( X >0) 2α P( X > α )] +CP( X > α ) 0, k 2α 1 P( X >k α )+1] This completes the proof of the theorem. Ackowledgmets. (38) The authors are most grateful to the Editor Bigge Zhag adaoymousrefereeforthecarefulreadigofthepaper ad valuable suggestios which helped i improvig a earlier versio of this paper. This work was supported by the Natioal Natural Sciece Foudatio of Chia ( , , ad ), the Specialized Research Fud for the Doctoral Program of Higher Educatio of Chia ( ), the Natural Sciece Foudatio of Ahui Provice ( M12, QA03), the Natural Sciece Foudatio of Ahui Educatio Bureau (KJ2010A035), the 211 proect of Ahui Uiversity, the Academic Iovatio Team of Ahui Uiversity (KJTD001B), ad the Studets Sciece Research Traiig Program of Ahui Uiversity (KYXL ). Refereces [1] J. Cuzick, A strog law for weighted sums of i.i.d. radom variables, Joural of Theoretical Probability, vol. 8, o. 3, pp , [2]Z.Bai,P.E.Cheg,adC.-H.Zhag, Aextesioofthe Hardy-Littlewood strog law, Statistica Siica, vol. 7, o. 4, pp , [3] Z. Bai ad P. Cheg, Marcikiewicz strog laws for liear statistics, Statistics & Probability Letters, vol. 46, o. 2, pp , [4] G.-H. Cai, Marcikiewicz strog laws for liear statistics of ρ -mixig sequeces of radom variables, Aais da Academia Brasileira de Ciêcias,vol.78,o.4,pp ,2006. [5] Q. Wu, A strog limit theorem for weighted sums of sequeces of egatively depedet radom variables, Joural of Iequalities ad Applicatios,vol.2010,ArticleID383805,8pages,2010. [6] S.H.Sug, Othestrogcovergeceforweightedsumsof radom variables, Statistical Papers, vol. 52, o. 2, pp , [7] X.-C. Zhou, C.-C. Ta, ad J.-G. Li, O the strog laws for weighted sums of ρ -mixig radom variables, Joural of Iequalities ad Applicatios, vol. 2011, Article ID , 8 pages, [8] X.Wag,X.Li,W.Yag,adS.Hu, Ocompletecovergece forarraysofrowwiseweaklydepedetradomvariables, Applied Mathematics Letters,vol.25,o.11,pp ,2012. [9] T. K. Chadra ad S. Ghosal, Extesios of the strog law of large umbers of Marcikiewicz ad Zygmud for depedet variables, Acta Mathematica Hugarica,vol.71,o.4,pp , [10] T. K. Chadra ad S. Ghosal, The strog law of large umbers for weighted averages uder depedece assumptios, Joural of Theoretical Probability,vol.9,o.3,pp ,1996. [11] Y. Wag, J. Ya, F. Cheg, ad C. Su, The strog law of large umbers ad the law of the iterated logarithm for product sums of NA ad AANA radom variables, Southeast Asia Bulleti of Mathematics,vol.27,o.2,pp ,2003. [12] M.-H. Ko, T.-S. Kim, ad Z. Li, The Háeck-Rèyi iequality for the AANA radom variables ad its applicatios, Taiwaese Joural of Mathematics,vol.9,o.1,pp ,2005. [13]D.YuaadJ.A, Rosethaltypeiequalitiesforasymptotically almost egatively associated radom variables ad applicatios, Sciece i Chia. Series A,vol.52, o.9,pp , [14] X. Wag, S. Hu, ad W. Yag, Covergece properties for asymptotically almost egatively associated sequece, Discrete Dyamics i Nature ad Society, vol. 2010, Article ID , 15 pages, [15] X.Wag,S.Hu,adW.Yag, Completecovergeceforarrays of rowwise asymptotically almost egatively associated radom variables, Discrete Dyamics i Nature ad Society, vol. 2011, Article ID , 11 pages, [16] X. Wag, S. Hu, W. Yag, ad X. Wag, O complete covergece of weighted sums for arrays of rowwise asymptotically almost egatively associated radom variables, Abstract ad Applied Aalysis,ArticleID315138,15pages,2012. [17] X.P.Hu,G.H.Fag,adD.J.Zhu, Strogcovergeceproperties for asymptotically almost egatively associated sequece, Discrete Dyamics i Nature ad Society, vol.2012,articleid , 8 pages, 2012.

7 Discrete Dyamics i Nature ad Society 7 [18] W. Yag, X. Wag, N. Lig, ad S. Hu, O complete covergece of movig average process for AANA sequece, Discrete Dyamics i Nature ad Society,vol.2012,ArticleID863931,24 pages, [19] W. Feller, A limit theorem for radom variables with ifiite momets, America Joural of Mathematics,vol.68,o.2,pp , 1946.

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