COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF ROWWISE END RANDOM VARIABLES
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1 GLASNIK MATEMATIČKI Vol. 4969)2014), COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF ROWWISE END RANDOM VARIABLES Yogfeg Wu, Mauel Ordóñez Cabrera ad Adrei Volodi Toglig Uiversity ad Soochow Uiversity, Chia, Uiversity of Sevilla, Spai ad Uiversity of Regia, Caada Abstract. The authors study complete covergece ad complete momet covergece for arrays of rowwise exteded egatively depedet END) radom variables ad obtai some ew results. The results exted ad improve the correspodig theorems by Sug 2005), Hu ad Taylor 1997), Hu et al. 1989), ad Chow 1988). 1. Itroductio The cocept of egatively orthat depedet NOD) radom variables was itroduced by Ebrahimi ad Ghosh [4]). Defiitio 1.1. The radom variables X 1,...,X k are said to be egatively upper orthat depedet NUOD) if for all real x 1,...,x k, PX i > x i,i = 1,2,...,k) k PX i > x i ), i=1 ad egatively lower orthat depedet NLOD) if PX i x i,i = 1,2,...,k) k PX i x i ). Radom variables X 1,...,X k are said to be NOD if they are both NUOD ad NLOD. i= Mathematics Subject Classificatio. 60F15. Key words ad phrases. Exteded egatively depedet radom variable, complete covergece, complete momet covergece. 449
2 450 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN The cocept of exteded egatively depedet END) radom variables was itroduced by Liu [11]). Defiitio 1.2. We call radom variables {X i,i 1} END if there exists a costat M > 0 such that both PX i x i,i = 1,2,...,) M PX i x i ) ad PX i > x i,i = 1,2,...,) M hold for each = 1,2,... ad all x 1,...,x. i=1 PX i > x i ), Clearly the END structure is substatially more comprehesive tha the NOD structure i that it ca reflect ot oly a egative depedece structure but also a positive oe, to some extet. Joag-Dev ad Proscha [10]) also poited out that egatively associated NA) radom variables must be NOD ad NOD is ot ecessarily NA, thus NA radom variables are END. Liu [11] also provided some iterestig examples to illustrate that the exteded egative depedece ideed allows a wide rage of depedece structures. Sice the article of Liu [11]) appeared, Che et al. [2]), Wu ad Gua [14]) ad Qiu et al. [12]) studied the covergece properties for END radom variables. Asequeceofradomvariables{U, 1}issaidtocovergecompletely to a costat a if for ay ε > 0, P U a > ε) <. I this case we write U a completely. This otio was give by Hsu ad Robbis [5]). Let {Z, 1} be a sequece of radom variables ad a > 0, b > 0, q > 0. If a E{b 1 Z ε} q < for some or all ε > 0, the the result was called the complete momet covergece by Chow [3]). I the followig we let {X k,1 k, 1} be a array of radom variables defied o a probability space Ω,F,P), {, 1} be a sequece of positive itegers such that lim =, ad {c, 1} be a sequece of positive costats such that c =. A array of rowwise radom variables {X k,1 k, 1} is said to be uiformly bouded by a radom variable X deoted by {X k } X) i=1
3 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 451 if there exists a costat C > 0 such that supp X k > x) CP X > x), for all x > 0.,k Clearly if {X k } X, for 0 < p < ad ay 1 k, 1, the E X k p CE X p. Hu et al. [7]) stated the followig complete covergece theorem for arrays of rowwise idepedet radom variables. Theorem 1.3. Let {X k,1 k, 1} be a array of rowwise idepedet radom variables ad {c, 1} be a sequece of positive costats such that c =. Suppose that for every ε > 0, some > 0 ad η 2, c P X k > ε) <, ad The k η c EXk 2 I X k )) < EX k I X k ) 0 as. ) c P X k > ε < for all ε > 0. The proof by Hu et al. give i [7] is mistakely based o the fact that the assumptios of Theorem 1.3 imply X k 0 i probability as. Hu ad Volodi [9]) foud that 1.5) does ot ecessarily follow from the assumptios of Theorem 1.3. Therefore, they replaced coditio c = by the coditio limif c > 0. I this case the assumptios of Theorem 1.3 imply 1.5). Sug [13]) proved Theorem 1.3 without the assumptio limif c > 0. Che et al. [1]) exteded Theorem 1.3 for the case of arrays of rowwise egatively associated radom variables. Hu ad Taylor [8]) proved the followig results. Theorem 1.4. Let {X k,1 k, 1} be a array of rowwise idepedet radom variables ad let {a, 1} be a sequece of positive
4 452 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN real umbers with a. Assume that Ψt) is a positive eve fuctio that satisfies Ψ t ) Ψ t ) t p ad as t t p1 for some iteger p 2. If ad EX k = 0,1 k, 1, EΨX k ) < Ψa ) Xk ) ) 2 2k E <, a where k is a positive iteger, the 1.7), 1.8), ad 1.9) imply 1 X k 0 a.s.. a Theorem 1.5. Let {X k,1 k, 1} be a array of rowwise idepedet radom variables ad let {a, 1} be a sequece of positive real umbers with a. If Ψt) is a positive eve fuctio that satisfies 1.6) for p = 1, the 1.7) ad 1.8) imply 1.10). I additio, Hu et al. [6]) obtaied the followig complete covergece. Theorem 1.6. Let {X k,1 k, 1} be a array of rowwise idepedet radom variables with 1.7) ad assume that {X k } X. If E X 2p < for some 1 p < 2, the 1/p X k 0 completely. Chow [3]) obtaied the followig complete momet covergece. Theorem 1.7. Suppose that {X, 1} is a sequece of idepedet ad idetically distributed radom variables with EX 1 = 0, α > 1/2, p 1 ad αp > 1. If E{ X 1 p X 1 log1 X 1 )} <, the { } αp 2 α ε α E X k < for all ε > 0. I this work, we shall exted ad improve Theorem 1.3 to END istead of idepedet or NA, ad shall exted ad improve Theorem uder some weaker coditios. It is worthy to poit out that we study complete momet covergece for the arrays of END radom variables uder some similar coditios, which were ot cosidered i Hu et al. [7]), Sug [13]) ad Che et al. [1]).
5 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 453 I the paper, C will deote geeric positive costats, whose value may vary from oe applicatio to aother, IA) will idicate the idicator fuctio of A. 2. Mai results We will preset the mai results of the paper ad the proofs will be detailed i the ext sectio. Theorem 2.1. Let {X k,1 k, 1} be a array of rowwise END radom variables ad let {c, 1} be a sequece of positive costats. Suppose that the followig coditios hold: i) for every ε > 0 P max j 1j i=1 ii) there exists η 1 ad > 0 such that ) a i X i > 1/p ε < for all ε > 0; k η c EXk 2 I X k )) <. The c P Xk EX k I X k ) ) ) > ε < for all ε > 0. Corollary 2.2. Let {X k,1 k, 1} be a array of rowwise END radom variables ad let {c, 1} be a sequece of positive costats. The 2.13), 2.14) ad 1.3) imply 1.4). Remark 2.3. Sice idepedece implies END ad we cosider η 1 istead of η 2, Corollary 2.2 exteds ad improves Theorem 1.3. I additio, compared with the results of Qiu et al. [12, Theorem 1]), Corollary 2.2 ad Theorem 1 of Qiu et al. [12]) do ot completely overlap with each other, although the coditios of our result have some similarities to those of Qiu et al. i [12]. Let c = 1, = for 1 ad let {a, 1} be a sequece of positive real umbers with a. Assumig that 1.7) holds ad replacig X k by X k /a i formulatio of Corollary 2.2, we ca obtai the followig corollary. Corollary 2.4. Let {X k,1 k, 1} be a array of rowwise END radom variables with 1.7) ad let {a, 1} be a sequece of positive real umbers with a. Suppose that the followig coditios hold:
6 454 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN i) for every ε > 0 P X k > a ε) < ; ii) there exists η 1 ad > 0 such that η EXk 2 I X k a )) < ; iii) The a 2 a 1 1 EX k I X k a ) 0. a X k 0 completely. Remark 2.5. The followig statemets show that the coditios of Corollary 2.4 are weaker tha those of Theorems 1.4 ad 1.5. Firstly, we state that 1.6)-1.8) imply 2.16). Without loss of geerality we may assume 0 < ε < 1. If p 2 or p = 1, by 1.6) ad 1.8), we have 2.1) = P X k > a ε) EI X k > a ε)) ε 1 E X k p1 a ε) p1 Ia ε < X k a ) ε p1) ε 1 ) E X k p I X k > a ) a p EΨX k ) Ψa ) <. E X k a ε I X k > a ε) Secodly, we take = 1 ad show that 1.6), 1.8) ad 1.9) imply 2.17). By 1.6) ad 1.8), we ca get easily η ) η a 2 EXkI X 2 EΨX k ) k a )) <. Ψa )
7 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 455 If p 2, take η = 2k, where k is a positive iteger. By 1.9), we ca get η 2k EXkI X 2 k a )) EXk) 2 <. a 2 Fially, we take = 1 ad show that 1.6)-1.8) imply 2.18). By 1.6)- 1.8), we have a 1 EX k I X k a ) = a 1 EX k I X k > a ) a 1 E X k I X k > a ) a 2 EΨX k ) Ψa ) 0 as. To sum up, we kow that Corollary 2.4 improve Theorems 1.4 ad 1.5. Obviously, complete covergece implies almost sure covergece. Therefore, our coclusios are much stroger ad coditios are much weaker. Takig a = 1/p for 1 p < 2 i Corollary 2.4, we ca obtai the followig corollary. Corollary 2.6. Let {X k,1 k, 1} be a array of rowwise END radom variables satisfyig 1.7). Suppose that the followig coditios hold: i) for every ε > 0 P X k > 1/p ε) < ; ii) there exists η > p/2 p) ad > 0 such that η 2/p EXkI X 2 k )) 1/p < ; iii) where 1 p < 2. The 1.11) holds. 1/p EX k I X k 1/p ) 0, Remark 2.7. The followig statemets show that the coditios of Corollary 2.6 are weaker tha those of Theorem 1.6. Firstly, by {X k } X ad E X 2p <, we have P X k > 1/p ε) C P X > 1/p ε) CE X 2p <.
8 456 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN Secodly, sice E X 2p < for 1 p < 2, we kow E X 2 <. Hece, by η > p/2 p) ad {X k } X, we have η 2/p EXkI X 2 k )) 1/p C 1 2/p)η E X 2 ) η <. Fially, by 1.7), {X k } X ad E X 2p <, we have 1/p EX k I X k 1/p ) 1/p E X k I X k > 1/p ) 1 2p E X k 2p 2 I X k > 1/p ) C 1 2p 1 E X 2p 0 as. To sum up, we kow that Corollary 2.6 exteds ad improves Theorem 1.6. The followig theorem shows that, uder some appropriate coditios, we ca obtai complete momet covergece for the array of rowwise END radom variables. Theorem 2.8. Let {X k,1 k, 1} be a array of rowwise END radom variables ad let {c, 1} be a sequece of positive costats. Suppose that 2.14) ad the followig coditios hold: i) for every ε > 0 c E X k I X k > ε) < ; ii) there exists η > 1 ad > 0 such that E X k I X k > /16η) 0 as. The { c E Xk EX k I X k ) ) } ε < for all ε > 0. Corollary 2.9. Let {X k,1 k, 1} be a array of rowwise END radom variables with 1.7). The coditios 2.14), 2.23) ad 2.24) imply { } ε c E X k < for all ε > 0.
9 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 457 Proof. Note that, from 1.7) ad 2.24), we ca get EX k I X k ) = EX k I X k > ) E X k I X k > ) 0 as. Theforeverygiveε > 0,whileissufficietlylarge, EX k I X k ) < ε. Therefore, by 2.25), we have { > c E Xk EX k I X k ) ) } ε > { } c E X k EX k I X k ) ε { } 2ε c E X k The proof is complete. Let c = 1, = for 1 ad let {a, 1} be a sequece of positive real umbers with a. Replacig X k by X k /a i formulatio of Corollary 2.9, we ca obtai the followig corollary. Corollary Let {X k,1 k, 1} be a array of rowwise END radom variables satisfyig 1.7) ad let {a, 1} be a sequece of positive real umbers with a. Suppose that 2.17) ad the followig coditios hold: i) for every ε > 0 a 1. E X k I X k > a ε) < ; ii) there exists η > 1 ad > 0 such that E X k I X k > a /16η) 0 as. The a 1 { } a 1 a E X k ε < for all ε > 0. Remark Wu ad Zhu [15]) discussed complete covergece ad complete momet covergece for arrays of rowwise NOD radom variables. The coditios i Wu ad Zhu [15]) are similar to those of Hu ad Taylor
10 458 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN [8]). By some similar argumets i Remark 2.5, we ca show that the coditios of Wu ad Zhu [15]) imply 2.16)-2.18), 2.27) ad 2.28). Here we omit the details. Sice NOD implies END ad the coditios i this paper are weaker tha those of Wu ad Zhu i [15], Corollary 2.4 ad 2.10 improve Theorem 1.1 ad 1.3 i [15] by Wu ad Zhu, respectively. Takig = ad c = αp 2, ad replacig X k by X k / α for 1 k i Corollary 2.9, we ca obtai the followig corollary. Corollary Let {X k,k 1} be a sequece of END radom variables with EX k = 0. Suppose that the followig coditios hold: i) for every ε > 0 αp 2 α E X k I X k > α ε) < ii) there exists η > max{1, αp 1 2α 1 } ad > 0 such that α E X k I X k > α /16η) 0 as ad η αp 2 2α EXk 2 I X k )) α <, where α > 1/2, p 1 ad αp > 1. The coditios 2.30)-2.32) imply 1.12). Remark The followig statemets show that the coditios of Corollary 2.12 are weaker tha those of Theorem 1.7. Firstly, we state the coditios of Theorem 1.7 imply 2.30). If p > 1, by E X 1 p <, we have αp 2 α E X k I X k > α ε) = αp 1 α E X 1 I X 1 > α ε) C C αp 1 α E X 1 Im α ε < X 1 m1) α ε) m=1 m= m E X 1 Im α ε < X 1 m1) α ε) αp 1 α m αp α E X 1 Im α ε < X 1 m1) α ε) m=1 E X 1 p Im α ε < X 1 m1) α ε) E X 1 p <. m=1
11 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 459 If p = 1, by E{ X 1 X 1 log1 X 1 )} <, we have αp 2 α E X k I X k > α ε) C m=1 m E X 1 Im α ε < X 1 m1) α ε) 1 1logm)E X 1 Im α ε < X 1 m1) α ε) m=1 CE X 1 C logme X 1 Im α ε < X 1 m1) α ε) m=2 CE X 1 C/α E{ X 1 log X 1 /ε)}im α ε < X 1 m1) α ε) m=2 C11/αlog1/ε))E X 1 C/α E{ X 1 log X 1 }Im α ε < X 1 m1) α ε) m=2 CE{ X 1 X 1 log1 X 1 )} <. Secodly, by E X 1 p < ad αp > 1, we have α E X k I X k > α /16η) /16η) 1 p αp E X k p I X k > α /16η) C 1 αp E X 1 p 0 as. Fially, we state the coditios of Theorem 1.7 imply 2.32). If p 2, from E X 1 p <, we kow EX1 2 <. By η > max{1, αp 1 2α 1 }, we have ) η αp 2 2α EXk 2 I X k α ) αp 2 2α 1)η EX1) 2 η <.
12 460 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN If 1 p < 2, by αp > 1 ad η > 1, we have αp 2 2α EXkI X 2 k α ) = ) η αp 2 2α 1)η EX1 2 I X 1 α ) ) η 2 p)η 1 αp 1)η 1) E X 1 p ) η <. To sum up, we kow that Corollary 2.12 exteds ad improves Theorem Proofs To prove mai results i this paper, we eed the followig lemmas. Lemma 3.1 [11]). If radom variables {X, 1} are END, the {g X ), 1} are still END, where {g ), 1} are either all mootoe icreasig or all mootoe decreasig. Lemma 3.2. Let {X, 1} be a sequece of END radom variables with mea zero ad 0 < B = EX2 k <. If S = X k, the there exists a costat M > 0 such that x P S x) P max X k y)2mexp 1k y x y log 1 xy ) ) B for x > 0,y > 0. Remark 3.3. Wu ad Gua [14]) established a similar coclusio, i which the term Pmax 1k X k y) was magified as P X k y). Here we omit the details of the proof. We first state the proof of Theorem 2.1. Proof. Let ε > 0 be give. Without loss of geerality, we may assume 0 < ε <. For ay 1 k, 1, we have c P Xk EX k I X k ) ) ) > ε c P Xk EX k I X k ) ) ) > ε, { X k > } c P Xk EX k I X k ) ) ) > ε, { X k }
13 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 461 c P X k > ) c P Xk I X k ) EX k I X k ) ) ) > ε =: I 1 I 2. By 2.13), we ca get I 1 <. To prove 2.15), it suffices to show I 2 <. Let The Y k = IX k < )X k I X k )IX k > ), Z k = IX k < )IX k > ). I 2 = ) ) c P Yk EY k Z k EZ k > ε ) ) c P Yk EY k > ε/2 ) ) c P Zk EZ k > ε/2 =: I 3 I 4. By Markov iequality ad 2.13), we have I 4 C c E For ay ε > 0, let { N 1 = : ) Zk EZ k C c P X k > ) <. } P X k > ε/6η) ε/24η), N 2 = N N 1. We kow ) ) c P Yk EY k > ε/2 N 1 N 1 c 24η/ε c P X k > ε/6η) <. The it suffices to show that N 2 c P k ) ) Yk EY k > ε/2 <. Let B = EY k EY k ) 2. Take x = ε/2, y = ε/2η ad η 1. By
14 462 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN Lemma 3.2, we have ) ) c P Yk EY k > ε/2 N 2 c P N 2 =: I 5 I 6. max Y k EY k > ε/2η ) 2C c 1k N 2 eb B ε 2 /4η For ay N 2, by P X k > ε/6η) < ε/24η) ad ε <, we ca get max EY k max E Y k 1k 1k = max 1k { E Xk I X k ε/6η) E X k Iε/6η < X k )P X k > ) } P X k > ) P X k > ε/6η)ε/6η ε/4η. Therefore, for ay N 2, we have I 5 c P max Y k > ε/4η ) sice Y k X k ) 1k N 2 c P X k > ε/4η ) <. by 2.13) ) Note that for ay N 2 P X k > ) P X k > ε/6η) < ε/24η). Note that 24η/ε P X k > ) < 1 if N 2. By C r -iequality, 3.33), 2.13) ad 2.14), we have I 6 C C C N 2 c B ) η C N 2 c N 2 c EY 2 k ) η ) η EXk 2 I X k ) C c EXk 2 I X k ) N 2 c C ε/24η) ) η 1 c P X k > ) <. ) η ) η ) η P X k > )
15 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 463 The proof is complete. Fially we state the proof of Theorem 2.8. Proof. Let S = Xk EX k I X k ) ) ad ε > 0 be give. Without loss of geerality, we may assume 0 < ε <. We have c E { S ε } = c P S ε > t ) dt = { c P S > εt ) dt 0 0 P S > εt ) } dt c P S > ε ) c P S > t ) dt = : I 7 I 8. To prove 2.25), it suffices to show that I 7 < ad I 8 <. Notig that 2.23) implies 2.13), by Theorem 2.1, we have I 7 <. The we prove I 8 <. Clearly P S > t ) = P S > t, ) { X k > t} P S > t, ) { X k t} P X k > t)p Xk I X k t) EX k I X k ) ) ) > t. The we have I 8 c P X k > t)dt c P Xk I X k t) EX k I X k ) ) ) > t dt = : I 9 I 10. By 2.23), we have I 9 c E X k I X k > ) <. The we prove I 10 <. Let Y k = tix k < t)x k I X k t)tix k > t), Z k = tix k < t)tix k > t),
16 464 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN we have P Xk I X k t) EX k I X k ) ) ) > t = P Yk EY k Z k EZ k EX k I < X k t) ) ) > t ) ) P Yk EY k Z k EZ k EX k I < X k t) > t. From 2.24), we kow k max t t 1 EX k I < X k t) max P X k > ) Therefore, while is sufficietly large, holds uiformly for t. Hece t t 1 E X k I < X k t) E X k I X k > ) 0 as. EX k I < X k t) < t/2 P Xk I X k t) EX k I X k ) ) ) > t ) ) ) P Yk EY k Zk EZ k > t/2 ) ) ) ) P Yk EY k > t/4 P Zk EZ k > t/4. The we have I 10 ) ) c P Zk EZ k > t/4 dt ) ) c P Yk EY k > t/4 dt = : I 11 I 12.
17 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 465 For I 11, by Markov iequality ad 2.23), we have I 11 C C c t 1 E Z k dt C c c E X k I X k > ) <. P X k > t)dt Next we cosider I 12. Let B = EY k EY k ) 2, x = t/4, y = t/4η ad η > 1. By Lemma 3.2, we have I 12 c C P max 1k Y k EY k > t/4η ) dt B ) ηdt c B t 2 /16η = : I 13 I 14. From 2.24), we kow that, while is sufficietly large, P X k > /16η) E X k I X k > /16η) < 1/32η. Hece, by 3.34), we have max t max t 1 EY k max max t 1 E Y k 1k t 1k max t { max t 1 E X k I X k /16η) 1k t 1 E X k I/16η < X k t)p X k > t) } max t 1/16η 1/16η2 { max t 1 /16η P X k > /16η)P X k > t) } 1k P X k > /16η) P X k > /16η) < 1/8η. P X k > )
18 466 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN Therefore, while is sufficietly large, we kow that max 1kk EY k < t/8η holds uiformly for t. Hece, by 2.23), we have I 13 c c c P max 1k Y k > t/8η ) dt sice Y k X k ) P max 1k X k > t/8η ) dt P X k > t/8η ) dt c E X k I X k > /8η ) <. Fially, we prove I 14 <. By C r -iequality, we have I 14 C = C c t 2 ) ηdt ηdt B C c t 2 EYk) 2 c t 2 EXk 2 I X k ) t 2 EXk 2 I < X k t) P X k > t)) ηdt C C C ηdt c t 2 EXk 2 I X k )) ηdt c t 1 E X k I < X k t)) ) ηdt c P X k > t) = : I 14 I By η > 1 ad 2.14), we have I 14 = C ) η c EXk 2 I X k ) t 2η dt C η c EXkI X 2 k )) <. 14 I 14.
19 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE 467 From 2.24), while is sufficietly large, we ca get E X k I X k > ) < 1. Hece, by η > 1 ad 2.23), we have I 14 C C ) η c E X k I X k > ) t η dt c E X k I X k > ) <. From 2.24), while is sufficietly large, we kow P X k > t) P X k > ) holds uiformly for t. Hece, by 2.23), we have I 14 C C The proof is complete. c P X k > t)dt E X k I X k > ) < 1 c E X k I X k > ) <. Ackowledgemets. The authors are extremely grateful to the referee for very carefully readig the mauscript ad for providig some substatial commets ad suggestios which eabled them to greatly improve the paper. The research of Y. Wu was supported by the Humaities ad Social Scieces Foudatio for the Youth Scholars of Miistry of Educatio of Chia 12YJCZH217) ad the Natural Sciece Foudatio of Ahui Provice MA03). Refereces [1] P. Y. Che, T. C. Hu, X. Liu ad A. Volodi, O complete covergece for arrays of rowwise egatively associated radom variables, Theory Probab. Appl ), [2] Y. Q. Che, A. Y. Che ad K. W. Ng, The strog law of large umbers for exteded egatively depedet radom variables, J. Appl. Probab ), [3] Y. S. Chow, O the rate of momet covergece of sample sums ad extremes, Bull. Ist. Math. Acad. Siica ), [4] N. Ebrahimi ad M. Ghosh, Multivariate egative depedece, Comm. Statist. ATheory Methods ), [5] P. L. Hsu ad H. Robbis, Complete covergece ad the law of large umbers, Proc. Nat. Acad. Sci. U. S. A ), [6] T. C. Hu, F. Móricz ad R. L. Taylor, Strog laws of large umbers for arrays of rowwise idepedet radom variables, Acta Math. Hugar ),
20 468 Y. WU, M. ORDÓÑEZ CABRERA AND A. VOLODIN [7] T. C. Hu, D. Szyal ad A. Volodi, A ote o complete covergece for arrays, Statist. Probab. Lett ), [8] T. C. Hu ad R. L. Taylor, O the strog law for arrays ad for the bootstrap mea ad variace, It. J. Math. Math. Sci ), [9] T. C. Hu ad A. Volodi, Addedum to A ote o complete covergece for arrays, Statist. Probab. Lett ), [10] K. Joag-Dev ad F. Proscha, Negative associatio of radom variables with applicatios, A. Statist ), [11] L. Liu, Precise large deviatios for depedet radom variables with heavy tails, Statist. Probab. Lett ), [12] D. H. Qiu, P. Y. Che, R. A. Giuliao ad A. Volodi, O the complete covergece for arrays of rowwise exteded egatively depedet radom variables, J. Korea Math. Soc ), [13] S. H. Sug, A. Volodi ad T. C. Hu, More o complete covergece for arrays, Statist. Probab. Lett ), [14] Y. F. Wu ad M. Gua, Covergece properties of the partial sums for sequeces of END radom variables, J. Korea Math. Soc ), [15] Y. F. Wu ad D. J. Zhu, Covergece properties of partial sums for arrays of rowwise egatively orthat depedet radom variables, J. Korea Statist. Soc ), Y. Wu College of Mathematics ad Computer Sciece Toglig Uiversity Toglig Chia ad Ceter for Fiacial Egieerig ad School of Mathematical Scieces Soochow Uiversity Suzhou Chia wyfwyf@126.com M. Ordóñez Cabrera Departmet of Mathematical Aalysis Uiversity of Sevilla Sevilla Spai cabrera@us.es A. Volodi Departmet of Mathematics ad Statistics Uiversity of Regia S4S 0A2 Saskatchewa Caada Adrei.Volodi@uregia.ca Received: Revised:
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