A^VÇÚO 1 32 ò 1 5 Ï 2016 c 10 Chese Joural of Appled Probablty ad Statstcs Oct., 2016, Vol. 32, No. 5, pp. 489-498 do: 10.3969/j.ss.1001-4268.2016.05.005 Complete Covergece for Weghted Sums of Arrays of Rowwse Asymptotcally Almost Negatve Assocated Radom Varables GE Meme School of Mathematcs ad Face, Chuzhou Uversty, Chuzhou, 239000, Cha WANG Xueju School of Mathematcal Sceces, Ahu Uversty, Hefe, 230601, Cha Abstract: Let {X, 1, 1} be a array of rowwse asymptotcally almost egatve assocated AANA, short radom varables. The complete covergece for weghted sums of arrays of rowwse AANA radom varables s establshed uder some geeral momet codtos. The result obtaed the paper geeralzes ad mproves the correspodg oe for egatvely assocated radom varables. Keywords: weghted sums; arrays of rowwse AANA radom varables; complete covergece 2010 Mathematcs Subject Classfcato: 60F15 1. Itroducto Throughout the paper, let IA be the dcator fucto of the set A. C deotes a postve costat whch may be dfferet varous places ad a = Ob stads for a Cb. Deote log x = l maxx, e, where l x s the ature logarthm. The cocept of complete covergece was troduced by Hsu ad Robbs [1] as follows: a sequece of radom varables {U, 1} s sad to coverge completely to a costat C f P U C > ε < for all ε > 0. I vew of the Borel-Catell lemma, ths mples that U C almost surely a.s.. The coverse s true f the {U, 1} are depedet. Hsu ad Robbs [1] proved that the sequece of arthmetc meas of depedet ad detcally dstrbuted..d. radom varables coverges completely to the expected value The project was supported by the Natural Scece Foudato of Ahu Provce 1508085J06, 1608085QA02, the Key Projects for Academc Talet of Ahu Provce gxbjzd2016005, the Scece Research Project of Chuzhou Uversty 2015GH35, the Qualty Egeerg Project of Ahu Provce 2015jyxm045 ad the Qualty Improvemet Project for Udergraduate Educato of Ahu Uversty ZLTS2015035. Correspodg author, E-mal: wxjahdx2000@126.com. Receved October 10, 2014. Revsed September 16, 2015.
490 Chese Joural of Appled Probablty ad Statstcs Vol. 32 f the varace of the summads s fte. Recetly, Ca [2] obtaed the followg complete covergece result for weghted sums of detcally dstrbuted egatvely assocated NA, short radom varables. Theorem 1 Let {X, X, 1} be a sequece of detcally dstrbuted NA radom varables, ad let {a, 1, 1} be a array of costats satsfyg A α = lm sup a α < for some 0 < α 2. Let b = 1/α log 1/γ for some γ > 0. Furthermore, suppose that EX = 0 whe 1 < α 2. If E exph X γ < for some h > 0, the 1 P max 1 m m a X > εb < for all ε > 0. Lookg at the momet codto E exph X γ <, t seems very strog. Ca t be replaced by weaker momet codto such as E X β < for some β > 0? The aswer s postve. I ths paper, we shall ot oly establsh Theorem 1 uder much weaker momet codto but also geeralze Theorem 1 for NA radom varables to the case of AANA radom varables. Frstly, let us recall the defto of AANA radom varables. Defto 2 A sequece {X, 1} of radom varables s sad to be asymptotcally almost egatvely assocated AANA, short f there exsts a oegatve sequece u 0 as such that Cov fx, gx +1, X +2,..., X +k u Var fx Var gx +1, X +2,..., X +k 1/2 for all, k N ad for all coordatewse odecreasg cotuous fuctos f ad g for whch Var fx ad Var gx +1, X +2,..., X +k exst. The famly of AANA radom varables cotas NA partcular, depedet radom varables wth u = 0, 1 ad some more kds of radom varables whch are ot much devated from beg egatvely assocated. A example of AANA radom varables whch are ot NA was costructed by Chadra ad Ghosal [3]. I addto, AANA s dfferet from asymptotcally egatvely assocated ANA, short radom varables, whch was troduced by Zhag ad Wag [4]. For varous results ad applcatos of AANA radom varables, oe ca refer to [3, 5 14] ad amog others. Our goal ths paper s to further study the complete covergece for weghted sums of arrays of AANA radom varables uder some weaker momet codtos. The results
No. 5 GE M. M., WANG X. J.: Complete Covergece for Weghted Sums 491 preseted ths paper are obtaed by usg the momet equalty of AANA radom varables ad the trucated method. Defto 3 A array {X, 1, 1} of radom varables s sad to be stochastcally domated by a radom varable X f for every, N there exsts a postve costat C such that P X x CP X x for all x 0. 2. Prelmary Lemmas I order to prove the ma result of the paper, we eed the followg mportat lemma. Lemma 4 [8] Let {X, 1} be a sequece of AANA radom varables wth u ad assume that {f, 1} are all odecreasg or all ocreasg ad cotuous fuctos, the {f X, 1} are stll AANA radom varables wth u. Lemma 5 [8] Let {X, 1} be a sequece of AANA radom varables wth EX = 0 for all 1 ad p 3 2 k 1, 4 2 k 1 ], where teger umber k 1. If u 1/p 1 <, the there exsts a postve costat D p depedg oly o p such that for all 1, E max S p [ p/2 ] D p E X p + EX 2. 1 The ext oe s the basc property for stochastc domato. For the proof, oe ca refer to [15], or [16]. Lemma 6 Let {X, 1} be a sequece of radom varables, whch s stochastcally domated by a radom varable X. The for ay α > 0 ad b > 0, the followg two statemets hold: E X α I X b C 1 {E X α I X b + b α P X > b}, where C 1 ad C 2 are postve costats. Lemma 7 [17] E X α I X > b C 2 E X α I X > b, Let {a, 1, 1} be a array of costats satsfyg a α = O for some α > 0. Let b = 1/α log 1/γ for some γ > 0. The CE X 1 α, α > γ; P a X > b CE X α log1 + X, α = γ; CE X γ, α < γ.
492 Chese Joural of Appled Probablty ad Statstcs Vol. 32 Lemma 8 [18] of costats satsfyg a = 0 or a > 1, ad Let X be a radom varable ad {a, 1, 1} be a array a α for some α > 0. Let b = 1/α log 1/α. If q > α, the 1 b q E a X q I a X b CE X α log1 + X. 3. Ma Result ad Its Proof I ths secto, let X,, N, be a array of rowwse AANA radom varables,.e., for every N, X, N, are AANA radom varables wth the same mxg coeffcet u ad let a,, N, be a array of real umbers. Our ma result s as follows. Theorem 9 Let 0 < α 2, γ > 0, {X, 1, 1} be a array of rowwse AANA radom varables whch s stochastcally domated by a radom varable X ad EX = 0 for 1 < α 2. Let {a, 1, 1} be a array of costats satsfyg A β = lm sup A β, <, where β = maxα, γ, ad b = 1/α log 1/γ. Let A β, = u 1/q 1 ad q 3 2 k 1, 4 2 k 1 ], where teger umber k 1. If CE X α <, α > γ; CE X α log1 + X <, α = γ; CE X γ <, α < γ, the for ay ε > 0, Proof 1 P max 1 j a β, <, q > max2, 2γ/α j a X > εb <. 1 Wthout loss of geeralty, we may assume that a 0 for all 1 ad 1. For fxed 1, defe for 1 that X = b Ia X < b + a X I a X b + b Ia X > b, It s easy to check that 1 P max 1 j T j = j X j a X > εb EX, j = 1, 2,...,.
No. 5 GE M. M., WANG X. J.: Complete Covergece for Weghted Sums 493 1 P max a X > b + 1 P max 1 j 1 j P a X > b + 1 P 1 1 C =: I + J. P a X > b + 1 P max 1 j max 1 j j X > εb T j > εb max 1 j T j > εb max 1 j j j EX EX By Lemma 7, t s easy to see I <. To prove J <, we wll frstly show that j max 0 as. 2 b 1 1 j EX Whe 1 < α 2, we have by EX = 0, Lemma 6 ad Markov s equalty that j max EX P a X > b + b 1 Ea X I a X > b b 1 1 j C Cb α P a X > b + b 1 E a X I a X > b a α E X α + Cb α a α E X α CE X α log α/γ 0 as. 3 Whe 0 < α 1, by Lemma 6 ad Markov s equalty we have that j max EX b 1 1 j C Cb α P a X > b + b 1 E a X I a X b P a X > b + Cb 1 a α E X α + Cb α [E a X I a X b + b P a X > b ] a α E X α I a X b CE X α log α/γ 0 as. 4 By 3 ad 4, we ca get 2. Hece J C 1 P max 1 j T j > εb /2. For fxed 1, t s easly see that {X EX, 1 } are stll AANA wth mea zero by Lemma 4. Hece, t follows from Markov s equalty, C r equalty ad Lemma 5 that J C 1 b q E max T 1 j j q
494 Chese Joural of Appled Probablty ad Statstcs Vol. 32 C C + C + C 1 b q 1 b q E X q + C 1 b q E X 2 q/2 E a X q I a X b 1 b q 1 =: J 1 + J 2 + J 3. By Lemmas 6 ad 7, we ca see that J 3 C q/2 Ea X 2 I a X b P a X > b 1 P a X > b <. For J 1 ad J 2, frstly, we cosder α = γ, where b = 1/α log 1/α. To prove J 1 <, we dvde {a, 1 } to three subsets {a : a log s }, {a : log s < a 1}, {a : a > 1}, where s = q α 1, q > 2. We ca get Notg that J 1 = C + C 1 b q + C 1 b q 1 b q =: J 11 + J 12 + J 13. we have by Lemmas 6 ad 7 that, : a log s E a X q I a X b :log s < a 1 : a >1 E a X q I a X b E a X q I a X b : a log s a α log sα, J 11 C 1 b q {E a X q I a X b + b q P a X > b } : a log s C 1 E b 1 a X α I a X b : a log s + C 1 P a X > b : a log s CE X α 1 b α a α + CE X α log1 + X : a log s CE X α 1 log 1 sα + CE X α log1 + X <.
No. 5 GE M. M., WANG X. J.: Complete Covergece for Weghted Sums 495 By Lemma 6, we get J 12 C C + C 1 b q 1 b q :log s < a 1 :log s < a 1 1 =: J 121 + J 122. Notg that s = q α 1 ad we ca see that J 121 C C 1 b q P a X > b } :log s < a 1 :log s < a 1 b q E X q I X 1/α log s+1/α {E a X q I a X b + b q P a X > b } a q E X q I a X b a α, a q E X q I X b log s = C E X q I 1 1/α log 1 s+1/α < X 1/α log s+1/α q/α log q/α = C E X q I 1 1/α log 1 s+1/α < X 1/α log s+1/α 1 q/α log q/α CE X α <. By Lemma 7, we ca get J 122 CE X α log1 + X, hece, J 12 <. Notg that J 13 CE X α log1 + X < by Lemma 8, we have proved that J 1 <. By C r equalty, Lemmas 6 ad 7, we get J 2 C { 1 b q [Ea X 2 I a X b + b 2 P a X > b ] C 1 b q C 1[ CE X α q/2 { } q/2 Ea X 2 I a X b + C E b 1 a X α] q/2 + CE X α log1 + X 1 1 log q/2 + CE X α log1 + X <. } q/2 P a X > b Whe α > γ, we have E X α <. By Lemmas 6 ad 7, we ca see that J 1 C 1 b q {E a X q I a X b + b q P a X > b }
496 Chese Joural of Appled Probablty ad Statstcs Vol. 32 C 1 b α a α E X α + C C 1 log α/γ + CE X α <. 1 P a X > b Whe α < γ, we have E X γ <. By Lemmas 6 ad 7 aga, we ca see that J 1 C 1 b γ a γ E X γ + C C γ/α log 1 + CE X γ <. 1 I order to prove J 2 <, we cosder the followg two cases. P a X > b If α < γ 2 or γ < α 2, otg that E X α <, we have by C r equalty, Lemmas 6 ad 7 that J 2 C 1 b αq/2 a α E X α q/2 + C 1 P a X > b C 1 log αq/2r + C 1 P a X > b <. If γ > 2 α or γ 2 > α, we get E X 2 <. By a α = O, we get max a α C. Whe k α, we ca see that 1 j a k = a α a k α C k α/α = C k/α. 5 Notg that q > max2, 2γ/α γ, we have by C r equalty, Lemmas 6 ad 7 ad 5 that J 2 C { } q/2 1 b q Ea X 2 I a X b + C C 1 log q/γ + CE X γ <. 1 P a X > b Therefore, J < follows from the statemets above. Ths completes the proof of the Theorem. Remark 10 X γ < s weakeed by Compared Theorem 9 wth Theorem 1, the momet codto E exph CE X α <, α > γ; CE X α log1 + X <, α = γ; CE X γ <, α < γ. 6
No. 5 GE M. M., WANG X. J.: Complete Covergece for Weghted Sums 497 I addto, the NA radom varables are exteded to AANA radom varables. Therefore, the result of Theorem 9 geeralzes ad mproves the correspodg oe of Theorem 1. Remark 11 Complete covergece for weghted sums of arrays of rowwse AANA radom varables has bee studed by may authors, such as [9, 14], ad so o. Compared Theorem 9 wth Theorem 2.1 of [9], the momet codto 6 s weaker tha E exph X γ < Theorem 2.1 of [9]. I addto, the codto o a array of real umbers {a, 1, 1} s weaker tha the correspodg oe of [14]. Refereces [1] Hsu P L, Robbs H. Complete covergece ad the law of large umbers [J]. Proc. Nat. Acad. Sc. U.S.A., 1947, 332: 25 31. [2] Ca G H. Strog laws for weghted sums of NA radom varables [J]. Metrka, 2008, 683: 323 331. [3] Chadra T K, Ghosal S. Extesos of the strog law of large umbers of Marckewcz ad Zygmud for depedet varables [J]. Acta Math. Hugar., 1996, 714: 327 336. [4] Zhag L X, Wag X Y. Covergece rates the strog laws of asymptotcally egatvely assocated radom felds [J]. Appl. Math. J. Chese Uv. Ser. B, 1999, 144: 406 416. [5] Chadra T K, Ghosal S. The strog law of large umbers for weghted averages uder depedece assumptos [J]. J. Theoret. Probab., 1996, 93: 797 809. [6] Wag Y B, Ya J G, Cheg F Y, et al. The strog law of large umbers ad the law of the terated logarthm for product sums of NA ad AANA radom varables [J]. Southeast Asa Bull. Math., 2003, 272: 369 384. [7] Ko M H, Km T S, L Z Y. The Hájeck-Rèy equalty for the AANA radom varables ad ts applcatos [J]. Tawaese J. Math., 2005, 91: 111 122. [8] Yua D M, A J. Rosethal type equaltes for asymptotcally almost egatvely assocated radom varables ad applcatos [J]. Sc. Cha Ser. A, 2009, 529: 1887 1904. [9] Wag X J, Hu S H, Yag W Z. Complete covergece for arrays of rowwse asymptotcally almost egatvely assocated radom varables [J]. Dscrete Dy. Nat. Soc., 2011, 2011Artcle ID 717126: 1 11. [10] Wag X J, Hu S H, Yag W Z, et al. O complete covergece of weghted sums for arrays of rowwse asymptotcally almost egatvely assocated radom varables [J]. Abstr. Appl. Aal., 2012, 2012Artcle ID 315138: 1 15. [11] Yag W Z, Wag X J, Lg N X, et al. O complete covergece of movg average process for AANA sequece [J]. Dscrete Dy. Nat. Soc., 2012, 2012Artcle ID 863931: 1 24. [12] She A T, Wu R C. Strog ad weak covergece for asymptotcally almost egatvely assocated radom varables [J]. Dscrete Dy. Nat. Soc., 2013, 2013Artcle ID 235012: 1 7. [13] She A T, Wu R C. Strog covergece for sequeces of asymptotcally almost egatvely assocated radom varables [J]. Stochastcs, 2014, 862: 291 303.
498 Chese Joural of Appled Probablty ad Statstcs Vol. 32 [14] Wag X H, She A T, L X Q. A ote o complete covergece of weghted sums for array of rowwse AANA radom varables [J]. J. Iequal. Appl., 2013, 2013Artcle ID 359: 1 13. [15] Wu Q Y. Probablty Lmt Theory for Mxed Sequece [M]. Bejg: Scece Press of Cha, 2006. Chese [16] She A T. O the strog covergece rate for weghted sums of arrays of rowwse egatvely orthat depedet radom varables [J]. RACSAM, 2013, 1072: 257 271. [17] Sug S H. O the strog covergece for weghted sums of radom varables [J]. Statst. Papers, 2011, 522: 447 454. [18] Sug S H. O the strog covergece for weghted sums of ρ -mxg radom varables [J]. Statst. Papers, 2013, 543: 773 781. AANA ÅCþ\ÚÂñ5 rr ²ÆêÆ 7KÆ, ², 239000 Æ S ŒÆêÆ ÆÆ, Ü, 230601 Á : {X, 1, 1} AANA ÅCþ. 3 2 Ý^ e, ïäaana Å Cþ\ÚÂñ5. Jí2ÚU?NACþƒAJ. ' c: \Ú; AANA; Âñ5 ã aò: O211.4