Research Article A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

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1 Joural of Probability a Statistics Volume 2, Article ID 849, 3 ages oi:.55/2/849 Research Article A Limit Theorem for Raom Proucts of Trimme Sums of i.i.. Raom Variables Fa-mei Zheg School of Mathematical Sciece, Huaiyi Normal Uiversity, Huaia 2233, Chia Corresoece shoul be aresse to Fa-mei Zheg, 632@hytc.eu.c Receive 3 May 2; Revise 25 July 2; Accete August 2 Acaemic Eitor: Ma Lai Tag Coyright q 2 Fa-mei Zheg. This is a oe access article istribute uer the Creative Commos Attributio Licese, which ermits urestricte use, istributio, a rerouctio i ay meium, rovie the origial wor is roerly cite. Let {X, X i ; i } be a sequece of ieeet a ietically istribute ositive raom variables with a cotiuous istributio fuctio F, af has a meium tail. Deote S i X i,s a i X ii M a<x i M a V 2 i X i X 2,whereM ma i X i, X / i X i,aa>is a fie costat. Uer some suitable coitios, we show that t T t a / /V e{ W / } i D,, as,wheret a S S a is the trimme sum a {W t ; t } is a staar Wieer rocess.. Itrouctio Let {X ; } be a sequece of raom variables a efie the artial sum S i X i a V 2 i X i X 2 for, where X / i X i. I the ast years, the asymtotic behaviors of the roucts of various raom variables have bee wiely stuie. Arol a Villaseñor cosiere sums of recors a obtaie the followig form of the cetral limit theorem CLT for ieeet a ietically istribute i.i.. eoetial raom variables with the mea equal to oe, log S log 2 N as.. Here a i the sequel, N is a staar ormal raom variable, a, stas for covergece i istributio i robability, almost surely. Observe that, via the Stirlig formula, the relatio. ca be equivaletly state as / S e 2N..2

2 2 Joural of Probability a Statistics I articular, Remała awesołowsi 2 remove the coitio that the istributio is eoetial a showe the asymtotic behavior of roucts of artial sums hols for ay sequece of i.i.. ositive raom variables. Namely, they rove the followig theorem. Theorem A. Let {X ; } be a sequece of i.i.. ositive square itegrable raom variables with EX, Var X σ 2 > a the coefficiet of variatio γ σ/. The, oe has S! /γ e 2N as..3 Recetly, the above result was etee by Qi 3, who showe that wheever {X ; } is i the omai of attractio of a stable law L with ie α, 2, there eists a umerical sequece A for α 2, it ca be tae as σ such that S! /A e Γ α /α L,.4 as, where Γ α α e. Furthermore, Zhag a Huag 4 etee Theorem A to the ivariace ricile. I this aer, we aim to stuy the wea ivariace ricile for self-ormalize roucts of trimme sums of i.i.. sequeces. Before statig our mai results, we ee to itrouce some ecessary otios. Let {X, X ; } be a sequece of i.i.. raom variables with a cotiuous istributio fuctio F. Assume that the right etremity of F satisfies γ F su{ : F < },.5 a the limitig tail quotiet lim F a,.6 F eists, where F F. The, the above limit is e ca for some c,, af or X is sai to have a thic tail if c, a meium tail if <c<, a a thi tail if c. Deote M ma j X j. For a fie costat a>, we say X j is a ear-maimum if a oly if X j M a, M, a the umber of ear-maima is K a : Car { j ; X j M a, M }..7 These cocets were first itrouce by Paes a Steutel 5, a their limit roerties have bee wiely stuie by Paes a Steutel 5, Paes a Li 6,Li 7, Paes 8, a Hu a Su 9. Now,set S a : X i I{M a<x i M }, i.8

3 Joural of Probability a Statistics 3 where, ω A, I{A}, ω / A, T a : S S a,.9 which are the sum of ear-maima a the trimme sum, resectively. From Remar of Hu a Su 9, we have that if F has a meium tail a EX /, the T a / EX, which imlies that with robability oe Car{ : T a, } is fiite at most. Thus, we ca reefie T a ift a. 2. Mai Result Now we are reay to state our mai results. Theorem 2.. Let {X, X ; } be a sequece of ositive i.i.. raom variables with a cotiuous istributio fuctio F, a EX, Var X σ 2. Assume that F has a meium tail. The, oe has t /V { T a t e } W i D,, as, 2. where {W t ; t } is a staar Wieer rocess. I articular, whe we tae t, it yiels the followig corollary. Corollary 2.2. Uer the assumtios of Theorem 2., oe has /V T a e 2N, 2.2 as,wheren is a staar ormal raom variable. Remar 2.3. Sice W / is a ormal raom variable with E 2 W E W EW, EW W y mi, y y y 2. y y 2.3 Corollary 2.2 follows from Theorem 2. immeiately.

4 4 Joural of Probability a Statistics 3. Proof of Theorem 2. I this sectio, we will give the roof of Theorem 2.. I the sequel, let C eote a ositive costat which may tae ifferet values i ifferet aearaces a mea the largest iteger. Note that via Remar of Hu a Su 9, we have C : T a /. It follows that for ay δ>, there eists a ositive iteger R such that P su C >δ R <δ. 3. Cosequetly, there eist two sequeces δ m δ /2 a R m such that P su C >δ m <δ m. 3.2 R m The strog law of large umbers also imlies that there eists a sequece R m such that su C R m. 3.3 m Here a i the sequel, we tae R m ma{r m,r m}, a it yiels P su C >δ m <δ m R m su C R m m. 3.4 The, it leas to P log C P V V P V log C, su R m C >δ m log C, su R m C δ m 3.5 : A m, B m,,

5 Joural of Probability a Statistics 5 a A m, <δ m. By usig the easio of the logarithm log 2 /2 θ 2, where θ, ees o <, we have that B m, P V log C, su R m C δ m P R m t log C V V P R m t log C V V V R m t P R m t log C V V V R m t P R m t log C V V R m t R m t log C, su C δ m R m C C 2, su C 2 δ m 2 θ C R m R m t C C 2 2 θ C I su C 2 δ m R m R m t C, su C >δ m R m : D m, E m,, 3.6 where θ,..., t are - -value a E m, <δ m. Also, we ca rewrite D m, as D m, P V V R m t log C C C V R m t C 2 2 θ C I su C 2 δ m. R m 3.7 Observe that, for ay fie m, it is easy to obtai by otig that V 2 V. R m t log C C as, 3.8

6 6 Joural of Probability a Statistics A if R m t, the we have C t 2 V 2 C t θ t 2 C V, 3.9 as.ifr m < t, the R m < t. Deote F m, : V R m C 2 I 2 θ C 2 su R m C δ m, 3. a, by observig that 2 / θ 2 4 2, the we ca obtai F m, C V R m C 2 C V R m S S a 2 C V R m 2 C S V R m S a 2 3. : H m, L m,. For ay ε>, by the Marov s iequality, we have P R m 2 >ε C 2 ε E S R m S C ε R m Var S Cσ2 ε 2 R m. 3.2 The, H m,. To obtai this result, we ee the followig fact: V 2 σ 2, 2 i X i X i Xi 2, as. 3.3 Iee, 2 i X i X Xi i Xi 2 X 2 i Xi 2 i 2 X i Xi 2 /

7 Joural of Probability a Statistics 7 Now, we choose two costats N>a<δ< such that P X >N <δ. Hece, i view of the strog law of large umbers, we have for large eough 2 X i Xi 2 / 2 X i Xi 2 I Xi / >N 2 X N 2 i I Xi >N / o N 2 P X >N o o, 3.5 which together with 3.4 imlies that 2 i X i X Xi V 2 2 i Xi 2 i, 3.6 as. Furthermore, i view of the strog law of large umbers agai, we obtai V 2 2 i X i X i Xi i Xi 2 2 σ 2, 3.7 as, where σ 2 Var X >. For L m,,byotigthats a /S Hu a Su 9, thus we ca easily get, as see S a S a S S, 3.8 as. The, for ay <δ <, there eists a ositive iteger R such that S a P su δ <δ. 3.9 R Cosequetly, coule with 3.8, we have P L m, >δ P P C V C V S a S a 2 >δ S a, su <δ S a P su δ R R >δ δ. 3.2

8 8 Joural of Probability a Statistics Clearly, to show L m,, as,itissufficiet to rove V S a. 3.2 Iee, combie with 3.7, we oly ee to show S a As a matter of fact, by the efiitios of S a a K a, we have M a K a <S a M K a I view of the fact M, we ca get from Hu a Su 9 that S a M K a, 3.24 a thus it suffices to rove M K a Actually, for all ε, δ >, a N large eough, we ca have that K a M P >ε P K a K a P M >ε M δ>ε,su <δ N P M su δ N Observe that if F has a meium tail, the we have M / M / log log / by otig that M / log /c 9, where c is the limit efie i Sectio. Thus it follows M P su δ, N 3.27

9 Joural of Probability a Statistics 9 as N. Further, by the Marov s iequality a the boue roerty of EK a from Hu a Su 9, we have K a P M δ>ε,su <δ N δ P K a >ε C δ ε EK a C δ ε C δ ε, 3.28 a, hece, the roof of 3.22 is termiate. Thus L m, comlete the roof, it is sufficiet to show that follows. Fially, i orer to Y t : C V t W, 3.29 a, coule with 3.2, we oly ee to rove Y t : V S t W. 3.3 Let t f, t > ɛ, H ɛ f t ɛ, t ɛ, S, t > ɛ, Y,ɛ t V ɛ, t ɛ. 3.3 It is obvious that ma t t W H ɛ W t su t ɛ t W, as ɛ Note that ma Y S t Y,ɛ t ma ɛ S, 3.33 t ɛ t ɛ V V

10 Joural of Probability a Statistics a the, for ay ɛ >, by the Cauchy-Schwarz iequality a 3.7, it follows that lim ɛ lim su P ma Y t Y,ɛ t ɛ lim t ɛ ɛ S lim su P ɛ ɛ lim ɛ lim su lim ɛ lim su V ɛ C E S ɛ C S /2 Var lim ɛ lim su ɛ C C lim lim su ɛ. ɛ 3.34 Furthermore, we ca obtai su ɛ t V ɛ S t S ɛ t S t S su ɛ t V ɛ ɛ ɛ S t S su V ɛ ɛ t V t t su S ɛ t V ɛ ma S 2 ɛ 2 t ɛ V su ɛ t C ma S V C i Xi. V C ma V i Xi 3.35 Therefore, uiformly for t ɛ,, we have V ɛ S V t ɛ S t W t o P ɛ o P, 3.36

11 Joural of Probability a Statistics where W t : S t t /V.NoticethatH ɛ is a cotiuous maig o the sace D,. Thus, usig the cotiuous maig theorem c.f., Theorem 2.7 of Billigsley, it follows that Y,ɛ t H ɛ W t o P H ɛ W t, i D,, as Hece, 3.32, 3.34, a 3.37 coule with Theorem 3.2 of Billigsley lea to 3.3. The roof is ow comlete. 4. Alicatio to U-Statistics AusefulotioofaU-statistic has bee itrouce by Hoeffig. LetaU-statistic be efie as U h X i,...,x im, m i < <i m 4. where h is a symmetric real fuctio of m argumets a {X i ; i } is a sequece of i.i.. raom variables. If we tae m ah, the U reuces to S /. Assume that Eh X,...,X m 2 <,alet h Eh, X 2,...,X m, Û m h X i Eh Eh. i 4.2 Thus, we may write U Û R, 4.3 where R H X i,...,x im, m i < <i m m H,..., m h,..., m h i Eh Eh. It is well ow cf. Resic 2 that Cov Û,R, i Var R, as. m Theorem 2. ow is etee to U-statistics as follows.

12 2 Joural of Probability a Statistics Theorem 4.. Let U be a U-statistic efie as above. Assume that Eh 2 < a P h X,...,X m >. Deote Eh > a σ 2 Var h X /. The, t m U m /mv { t e } W, i D,, as, 4.6 where W is a staar wieer rocess, a V 2 i X i X 2. I orer to rove this theorem, by 3.7, we oly ee to rove m m m U t σ W, i D,, as. 4.7 If this result is true, the with the fact that m U Eh euce from E h < see Resic 2 a 4.3, Theorem 4. follows immeiately from the metho use i the roof of Theorem 2. with S / relace by m U. Now, we begi to show 4.7. By 4.3, we have m m m m U m Û m m m R m. 4.8 By alyig 3.3 to raom variables mh X i for i, we have m m Û m i h m i i h i t σ W, 4.9 i D,,as, sice the seco eressio coverges to zero as. Therefore, for rovig 4.7, we oly ee to rove m m R m, as, 4. a it is sufficiet to emostrate R : m R m, as. 4. m Iee, we ca easily obtai E R 2 as from Hoeffig. Thus, we comlete the roof of 4.7, a, hece, Theorem 3. hols.

13 Joural of Probability a Statistics 3 Acowlegmet The author thas the referees for valuable commets that have le to imrovemets i this wor. Refereces B. C. Arol a J. A. Villaseñor, The asymtotic istributios of sums of recors, Etremes, vol., o. 3, , G. Remała a J. Wesołowsi, Asymtotics for roucts of sums a U-statistics, Electroic Commuicatios i Probability, vol. 7, , Y. Qi, Limit istributios for roucts of sums, Statistics & Probability Letters, vol. 62, o.,. 93, L.-X. Zhag a W. Huag, A ote o the ivariace ricile of the rouct of sums of raom variables, Electroic Commuicatios i Probability, vol. 2,. 5 56, A. G. Paes a F. W. Steutel, O the umber of recors ear the maimum, The Australia Joural of Statistics, vol. 39, o. 2, , A. G. Paes a Y. Li, Limit laws for the umber of ear maima via the Poisso aroimatio, Statistics & Probability Letters, vol. 4, o. 4, , Y. Li, A ote o the umber of recors ear the maimum, Statistics & Probability Letters, vol. 43, o. 2, , A. G. Paes, The umber a sum of ear-maima for thi-taile oulatios, Avaces i Alie Probability, vol. 32, o. 4,. 6, 2. 9 Z. Hu a C. Su, Limit theorems for the umber a sum of ear-maima for meium tails, Statistics & Probability Letters, vol. 63, o. 3, , 23. P. Billigsley, Covergece of Probability Measures, Wiley Series i Probability a Statistics: Probability a Statistics, Joh Wiley & Sos Ic., New Yor, NY, USA, 2 eitio, 999. W. Hoeffig, A class of statistics with asymtotically ormal istributio, Aals of Mathematical Statistics, vol. 9, , S. I. Resic, Limit laws for recor values, Stochastic Processes a their Alicatios, vol., , 973.

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