Hájek-Rényi Type Inequalities and Strong Law of Large Numbers for NOD Sequences

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1 Appl Math If Sc 7, No 6, Appled Matheatcs & Iforato Sceces A Iteratoal Joural Háje-Réy Type Iequaltes ad Strog Law of Large Nuers for NOD Sequeces Ma Sogl Departet of Matheatcs, Chao Hu Uversty, Chaohu, 38000, HeFe, P R Cha Receved: 5 Apr 03, Revsed: 8 Aug 03, Accepted: 9 Aug 03 Pulshed ole: Nov 03 Astract: I the paper, we get the precse results of Háje-Réy type equaltes for the partal sus of egatvely orthat depedet sequeces, whch prove the results of Theore 3 ad Corollary 3 K 006 I addto, the Marcewcz type strog law of large uers s otaed At last, the strog stalty for weghted sus of egatvely orthat depedet sequeces s dscussed Keywords: Háje-Réy equalty; egatvely orthat depedet sequeces; strog law of large uers Itroducto We use the followg otatoslet X, e a sequece of rado varales defed o a fxed proalty space Deote S = = X ad T = = X EX for each ad IA e the dcator fucto of the set A Háje ad Réy 955 proved the followg portat equalty If X, s a sequece of depedet rado varales wth ea zero, ad, s a odecreasg sequece of postve real uers, the for ay ε > 0 ad postve teger <, X P ax = ε ε EX j + EX j j= j=+ j I the paper, we wll further study the Háje-Réy type equalty ad the strog law of large uers for egatvely orthat depedet sequeces Defto A fte collecto of rado varales X,X,,X s sad to e egatvely upper orthat depedet NUOD, f for all real uers x,x,,x, PX > x,=,,, PX > x, ad egatvely lower orthat depedet NLOD, f for all real uers x,x,,x, = PX x,=,,, PX x 3 A fte collecto of rado varales X,X,,X s sad to e egatvely orthat depedet NOD f they are oth NUOD ad NLOD = A fte sequece X, s sad to e NOD NUOD Or NLOD,f every fte sucollecto s NOD NUOD Or NLOD Lea cf Bozorga et al, 996 Let X, e a sequece of NOD rado varales, f, f, e all odecreasg or all ocreasg fuctos, the f X, s stll a sequece of NOD rado varales Lea cf K, 006Let X,X,,X e NOD rado varales wth EX = 0 ad EX < for all The we have p E X + p EX+ 4 = = for all tegers, p ad + p Moreover, we have E ax X log 3 + EX 5 = = By Lea ad Lea, we ca get the followg Corollary Corollary Khtche-Kologorov theore Let X, e a sequece of NOD rado varales If VarX log < 6 the X EX coverges as Correspodg author e-al: 7879@qqco c 03 NSP Natural Sceces Pulshg Cor

2 50 M Sogl: Háje-Réy Type Iequaltes ad Strog Law Háje-Réy type equaltes for NOD sequeces I ths secto, we wll gve Háje-Réy type equaltes for NOD sequeces, whch prove the results of K 006 Theore Let X, e a sequece of NOD rado varales ad, e a odecreasg sequece of postve uers The for ay ε > 0 ad ay teger, P ax j= 4 ε log 3 + ε j= j Proof Wthout loss of geeralty, we assue that for all Let α = For 0, defe A =:α < α + For A /0, we let v = ax : A ad t e the dex of the last oepty set A Ovously, A A j = /0 f j ad t A =,,, It s easly see that α =0 v < α + f A ad X EX, s also a sequece of NOD rado varales y Lea By Marov s equalty ad 6 Lea, we have P ax t =0,A /0 j= = P ax ax 0t,A /0 A j= t P =0,A /0 α ax v ε α E ax ε t =0,A /0 ε log 3 + ε ε ε j= v j= α log 3 v+ j= Now we estate v j= t =0,A /0,v j t α =0,A /0,v j α Let 0 = : A /0,v j, the j v0 < α 0+ follows fro the defto of v Therefore, t =0,A /0,v j α < = 0 = α α α 0 < α = 4 3 α j j Thus follows fro ad 3 edately Theore Let X, e a sequece of NOD rado varales ad, e a odecreasg sequece of postve uers The for ay ε > 0 ad ay postve tegers <, 4 ε P ax j= ε + 4[log j= 3 +] j=+ j Proof Oserve that ax thus j= + ax + P ax j=+ j= j=, ε 4 P ε j= + P ax + j=+ ε By Marov s equalty, Lea ad Theore,4 ca e otaed 3 Marcewcz type strog law of large uers for NOD sequeces Theore 3 Let X, e a sequece of detcally dstruted NOD rado varales wth P X /p log < for 0< p< Assue that EX = 0 f p<,the X /p 0 as, 3 = Deote Proof Y = /p IX /p +X I X < /p + /p IX /p the PX Y = Borel-Catell lea Thus /p P X /p <, whch ples that PX Y, o = 0 y the X 0 as f ad = oly f /p Y 0 as So we eed to show that = c 03 NSP Natural Sceces Pulshg Cor

3 Appl Math If Sc 7, No 6, / wwwaturalspulshgco/jouralsasp 5 ad /p /p Y EY 0 as,, 3 = = EY 0, 33 By Corollary ad Kroecer s lea, to prove 3, t suffces to show that I fact, Var Y /p Y Var /p log < 34 log EY C P X /p log + C C+C log /p C = = = /p log EX I X p < EX I X < /p log /p log /p E X p X p I X p < C+C /p E X p p/p I X p < = < Hece 3 holds Next, we wll prove 33 It wll e dvded to two cases: If p =, y E X p < ad Leesgue doated covergece theore, we have l /p P X /p =0, 35 l EX I X < /p = l Ω X ωi X ω < /p Pdω = EX = 0 36 Thus, EY /p P X /p + EX I X < /p 0, as By the Toepltz lea, we ota l EY = 0 = If p, y the Kroecer s lea, to prove 33, t suffces to show that For 0< p<, EY /p C+C j= C+ EY /p < 37 P X /p + j= j= = j E X I X < /p /p /p E X I j X p < j /p E X I j X p < j j /p E X p j p/p I j X p < j< For p<, y EX = 0, we ca see that EY /p P X /p + C+ C+C j= j= j j= EX I X < /p /p /p E X I X /p /p E X I j X p < j+ /p E X I j X p < j+ j /p E X p j p/p I j X p < j+ < Thus 37 holds, whch ples 33 y Kroecer s lea We get the desred result 4 Strog stalty for weghted sus of NOD sequeces I ths secto, we wll study the strog stalty for weghted sus of NOD rado varales Frstly, we wll gve soe deftos as follows: Defto 4 A rado varale sequece X, s sad to e stochastcally doated y a rado varale X f there exsts a costat C, such that P X >x CP X >x 4 for all x 0 ad Defto 4 A rado varale sequece Y, s sad to e strogly stale f there exst two costat sequeces, ad d, wth 0<, such that Y d 0 as 4 Lea 4 Let X, e a sequece of rado varales whch s stochastcally doated y a rado varale X For ay α > 0 ad > 0, the followg stateet holds: E X α I X CE X α I X + α P X > Where C s a postve costat Theore 4 Let a, ad, e two sequeces of postve uers wth c = /a, c = /a log for ad Let X, e a sequece of NOD rado varales whch s stochastcally doated y a rado varale X Defe Nx= Card : c x, Rx= x Nyy 3 dy, x>0 If the followg codtos are satsfed: Nx< for ay x>0, R= Nyy 3 dy<, EX R X <, the there exst d R,,, such that = a X d 0 as 43 c 03 NSP Natural Sceces Pulshg Cor

4 5 M Sogl: Háje-Réy Type Iequaltes ad Strog Law Proof Deote X c = c IX < c +X I X c +c IX > c,, thex c, ada X c /, are stll NOD fro Lea Sce Nx s odecreasg, the for ay x>0 Rx Nx x y 3 dy= x Nx, 44 whch ples that EN X EX R X < Therefore = P X X c = P X >c = C P X >c CEN X < 45 = By Borel-Catell lea for ay sequece d, R, the sequeces = sae set a X c = a X d ad d coverge to the sae lt o the We wll show that = a X c whch gves the theore wth d = It follows fro Lea 4 that Vara X c log EX c 0 as, = EX c a EX c 3 P X >c +3 EX I X c C P X >c +C EX I X c CEN X +C EX I X c 46 EX I X c = EX I X c :c + EX I X c = I + I 47 :c > Sce N = Card : c R < fro 44 ad codto, the I < I = EX I X c :c > = = <c EX I X c N N EX I X = N N EX I X = + N N EX I< X = = I + I I C N N = = C j j= j+ 3 = j= N N j 3 C j+ 3 N j+ C y 3 Nydy< j= Sce Nx s odecreasg ad Rx s ocreasg, the I = = = = = N N EX I< X N N EX I < X = = EX I < X C = C C Therefore EX I < X EX I < X N N N + = Nxx 3 dx REX I < X EX R X I < X CEX R X < Vara X c = log < 48 followg fro the aove stateets By Corollary ad Kroecer s Lea, t follows that a X c EX c 0 as 49 Tag d = = a EX c,, the = We coplete the proof of the theore Acowledgeet a X c d 0 as Ths wor s supported y Foudato of Ahu Educatoal Cottee KJ03Z5 The author s grateful to the aoyous referee for a careful checg of the detals ad for helpful coets that proved ths paper Refereces [] Bozorga, A, Patterso, R F, Taylor, R L, Lt theores for depedet rado varales World Cogress Nolear Aalysts, 9, [] Chrstofdes, T C, Maxal equaltes for deartgales ad a strog law of large uers Statst Proa Lett, 50, [3] Fazeas, I, Klesov, O, A geeral approach to the strog law of large uers Theory Proa Appl, 45, [4] Ga, S X, The Háje Réy equalty for Baach space valued artgales ad the p soothess of Baach space Statst Proa Lett, 3, [5] Háje, J, Réy, A, A geeralzato of a equalty of Kologorov Acta Math Acad Sc Hugar, 6, c 03 NSP Natural Sceces Pulshg Cor

5 Appl Math If Sc 7, No 6, / wwwaturalspulshgco/jouralsasp 53 [6] Hu, S H, Che, G J, Wag, X J, O extedg the Bru-Prohorov strog law of large uers for artgale dffereces Statst Proa Lett, 78, [7] Hu, S H, Wag, X J, Yag, W Z, Zhao, T, The Háje-Réy-type equalty for assocated rado varales Statst Proa Lett, 79, [8] Joag-Dev, K, Proscha, F, Negatve assocato of rado varales wth applcatos A Statst,, [9] K, H C, The Háje-Réy equalty for weghted sus of egatvely orthat depedet rado varales It J Cotep Math Sc,, [0] Lu, J J, Ga, S X, Che, P Y, The Háje-Réy equalty for NA rado varales ad ts applcato Statst Proa Lett, 43, [] Praasa Rao, B L S, Háje-Réy-type equalty for assocated sequeces Statst Proa Lett, 57, [] Sug, H S, A ote o the Háje-Réy equalty for assocated rado varales Statst Proa Lett, 78, Ma Sogl receved the MS degree Statstcs ad Proalty fro Ahu Uversty 007 He s curretly a teacher Chaohu Uversty Hs research terests are the areas of Proalty lt theores c 03 NSP Natural Sceces Pulshg Cor