在最少条件下的对数律精确渐近性

在最少条件下的对数律精确渐近性 应用概率统计 第二三期2006 第二十二卷 年8月 PreciseRates under ChineseJournalofAppliedPI'obability andStatisticsVo1.22No.3Aug.2006 intheLawoftheLogarithm MinimalConditions* ZHANGLIXINLINZHENGYAN (Depar?tmentofMathematics,ZhejiangUniversity,Hangzhou,310028) Abstract LetJYl,.Y2..一bei.i.d.randolnvaxiables,andsetS,I=l+???+I=m. a,xlSkl, n>1.ThepreciseratesinthelaxvofthelogaxithmforaxeobtainedunderSUfficientand necessaryconditions. Keywords:Tailt)I.obabilitiesofSUNISofi.i.d.randomvariables,thelawofthelogal'ithm, strongapproximation. AMSSubjectClassification:Primary60F15;Secondary60G50. ?1.IntroductionandMainResults Let{,,;?1)beasequenceofi.i.drandomvariableswithacommondistl'ibution functionF,andsetS,=?,^l=m.,axIsI,礼1.Alsoletlogx=ln(xVe)and()=1.一1n 瓣.Lai(1974)andChowandLai(1975)considerthefollowingresultonthelawofthe logarithm. TheoremASupposetha.tVarX=盯.andr1.Thenthefollowingareequivalent P(McO(n))<? E(n))<?, forallE>盯 f0rallE>盯-二; E(n))<?,forsomeE>0; EX=0andElXl/(1ogIxI)<oo Fol'7?=1,GutandSpgttaru(2000)gavethefollowingpreciseasymptotics TheoremBSupposethatEX=0andEX=<...Then,for051, 一量礼一(1ogn)aP(ISIlime26+2logP(IS礼_.f\o,. ?e~/—nlo—gn)=, +1'' where(+)isthe(25+2)一thabsolutemomentofthestandardnormaldistribution. Thepul'poseofthisimpel'istoCO1raiderthepreciseasymptoticsforallr>1.Weobtainthe sufficientandnecessaryconditionsforsuchkindofresultstohold.Hereareourmain1.esults. Researchsuppo,'tedbyNationalNatul'alScienceFoundationofChina(No.10471126) Received2006.4.7. 儿 ?? S ,lP ? 一 Z 7 ?? S ,lP ? 一 7 ?? 3112应用概率统计 The.rem1.1Let7.>1aI1d.>一1/2andlet.n(E)beafuncti.n.fEsuchthat an(F)log7-47.as佗-??andE\f S1ppose{)isase(11eI1ceofnon—negatiVenilmberssatisfying n" = ?^,?(1og).,n-?.. =1=1 ThenthefollowingaIeequivalent E=0 2 E.=盯(0<盯<O0)and ,F E[IXl(1ogIXI)一】<oo; nr一P{Mn盯(n)(E+0n(E))) exp卜27-)r(0+1/2),仃>0; ). li111fF7?一1)】叶/? (\v/, , nr一.,1P{ISI(n)(e+0n(E))) exp{一27.r(0+1/2),盯>0 r一-.,『】PIME盯))<?ifand.nlyifE>; t=J 一P{IS,IE盯))<..ifand.nlyifE> f::1 EX:0.EX=(0<仃<..)and li】一 一 (7—1)?————————一X (\1v'一1I E[IXl(1ogIX1)1<? ?tr一2P{n以E咖(n)十an}= lira一 @—2_(—7"--1)量P{It,,v,'一1n1 E盯(n)+an)= 2 盯>0: 第二十二卷 (1.8) (1.9) >0.(1.10) Als., T byThe l oIem1, I 1, ja in l】e (1.9)and(1.10)the ( widestanisalsoo(V@logn). .一一一,1: , Remarkum' 一. ' .dw.k3/2dMn.b. <.vilsi 1.1 T— ZhangandBaek ? 2003)andS o p d S~ s taru(200 w 4 o ) rK est o a r b ' l ' ishe d d / (1 a .10 u )to n r 3/2bingtheBe1YEsseeninequalityTheirmethdonot'. 工J usingad:ffe1.entappI..ach,HuangandZhang(2005)pr.vedaresultsimiar.(?5)胁.m ?2.NormalCases Intl1isse'ti.n,wep1..veTI1e.1.eIn1.1inthecaSethat{,1;n1)arenormalrand.m vm.iables.Letfw);t0eastandardwieneI'pr.cessand?astandardn.rmalvariabl.? ?? + 一 一 .一 l a '暑 m 出T .一n 0 一 :一一\ = 缸 IC>?Lr m=1 rh .二.薹 第三期张立新林正炎:在最少条件下的对数律精确渐近性313 Proposition2.1Letr>1anda>一1/2andletan(E)beafunctionofEsatisfying(1.1) Andlet{,t1)satisfy(1.2).Then and lim『F2一 \| 2 , o. 一 1)】卅/?n^P{sup n=10<s<1 exp{一2T)r(0+1/2) lim『F2一 e\? o. 一 1)】a+l/? n=1 (s)IX/,2logn(e) nr-2AP{INIx/~logn(e+an(E))) exp{一2T,/)r(0+1/2)(2.2) ProofByusingthetailprobabilitiesofanormalrandomvariableandWienerprocess ,the proofissimilm'tothatofProposition2.1ofHuangandZhang(2006)fc.f.Zhang(2001)).The detailsareonfittedhere. ?3.Approximation ThepurposeofthissectionistousethestrongapproximationandFeller'S(1945)andEin. mahl'S(1989)truncationmethodstoshowthattheprobabilitiesin(1.4)fortcanbeapproxi. matedbythosefol'sup1(s)Iandthepl'obabilitiesin(1.5)forcanbeapproximatedby u1 thoseforN. SupposethatEX=0andEX.=盯.<?.Withoutlosingofgenerality. weassumethat 盯=1throughoutthissection.LetP>1/2.Foreachnand1Jn,welet = 川Iv~/(1ogn)), S1j -nj 1 lll j | n{=Xj -- I J= 薹=II,=l二? = I(x/n/(1ogn)<II(n)), = XjI(II>(n)), 一 E【】 :Var( 七=l =x5一E[X/[j】 =一 E【】 Andalsodefine,,,, -- ,,Iand--IIsimilarly. Thefollowingpropositionisthemainresultofthissection. Proposition3.1Letr>1,a>--1and2P>P>1/2.Supposethatthecondition (1.3)issatisfied.Andlet{^)satisfy(1.2).Thenthereexist>0andasequenceofpositive numbers{口Jt)suchthat PfsupL0<<1 P{E(n))P{sup 0<s<1 +l ogn)P)_口.r 3 (1ogn)p)+口(3.1) , L一 ? = .J,.J—— 314应用概率统计 P{INIex/—2lo—gn+3(1ogn)pg, P{IS,Ie咖(7))P{INIEx/—2lo—gn forallt:?(,二一._:1+c),n1,alld 3 (1ogn)p)+‰ 第二十二卷 (3.2) (3.3) Toshowthisproposition,weneedsomelemmas. Lemma3.1Forallysequenceofindependentrandomvariables{?,1;n1)withmean zel?oandfinite,,ariancethel'eexistsasequenceofindependentnormalvariables{叼,;几1}with Eq,l=0andET=Efsuchthat,fo,allQ>2andY>0, Pfmaxl\<tl (AQ)Qy wheneverEIfI'<.C.i=1,???,n.Here,J4isauniversalconstant ProofSeeSakhanel~o(1980,1984,1985).社 Lemma:j.2LetQ2,fI,,…,lbeindependentrandomvariableswithE=0a,nd El&IQ<O0,k=1,…,礼Thenforall>0 Pfmaxk<" )f2exp{)+c2QQ一Q耋E-已-Q where.4isaunix,el'sa1(:onstantasinLemma3.1 ProofItfollowsfl'omLemma3.1easily.SeealsoPetrov(1995,Page78).社 Lemma3.3Deftne?,=I--s!,^一SI.Letr condition(1.3)issatisfiedandEX=1.Thenforany suchthat >1.a>--1andP>1/2.Supposethatthe >0thereexistaconstantK=KO',a,) (3.4) where厶=P(A,x/~,/(1ogn),--n!(礼)). ProofLet.=nE[IXII{IXl>元/(1ogn)P}].ThenlEEl,,1Jn.SettingIJII一11 wehave C=,击v/~/(1.gl.gn)) t1 {?,v/n,/(1oglogn))CU{xj?J),n?c J=1 S.f01.几?c,P(Xj?j,--,!(n)).Observerthat』=0whel1evel?_?j J1 ? < ,J p p 0 7 ,【 <一 g ? 一 x? Q ? E ?? Q <一 ,l,/ >一 叩 ? 一 ? ? ? ? ? < <一 , ? 一 礼 ?? 第三期张立新林正炎: 在最少条件下的对数律精确渐近性 ?礼,sothatwe11aVefo1'7largeen.ughandall1 礼, P(xj?j,A(n)) =P?m …ax131JVmax一I) :P(_YJ?(龄阮IVJ--/一J?) P(?)P(?(礼)一ff) P(Ixl>v~T/(1.g礼))p, -- … I (n)一v~/(1.g礼)) P(/xl>v~/(1.P)).A.taightfol'Wal'dapplicati.n.ftheine qllalities.fOttavianiandBernstein . P(丽:))2P(IS'I))礼 f0I'sortie卵()>0Itfollowstt1at )0 ?tl=f耋P(<IXl< 砉Pf,(1ogj)P<Ixl< AP(IX[> (1og(j十1))p (1og(j+1))p 善P(<IXl_< ,— (1og—n)p1 {n )圭礼一^n=1 (1og(j十1)) yields 1圭礼一(1ogn).tl=1 言P(<)). cE[IXI'一(1ogIX1)p()+n]<.o. 佗CZ.,f,henwehayeP(~-71咖(礼))Tt--~?Itloll.wsthat ,岳礼】厶,8,nr--S/2--r/A(1.g礼) 8 , 一.A(1.gn)E[IXII{IXI>V~/(I))】 8 , 叫(1ogn).黑,{<IXl.EI,{,.<<儿=l一,I.I『Infr,口'?-— = 妻{(1ogj)P8Ej- _ 1lOg =?JJJ{L CEE[IXlI1{PI{=LL【log.7J <JJ <ff (1og(j+1))p (1og(j+1))p (1og(j+1))p)] )]nr-3/2-u,n(1ogn) )】nr-3/2-o( (byLemma2.3ofHuangandZhang(2006)) 一 ,善,{<,~-a/2-,7 E[Ixl2(t-,1)(1.glX1)2p(.,,-1/2一)卅]<.o. 315 ?? <, 2 一 札 ? 316应腑概率统计第二十二卷 (3.4)isproved Lemma3.4Let7?>1,a>一 1andP>1/2.Supposethecondition(1.3)issatisfied.Then oo fol?any>0thel?eexistaconstantK=(r,0,P,)suchthat?n~-2f,lII,1K<..,where 儿=1 厶=P(??v~,/(1oglogn).,^?(n)). ProofObviously. P(/(1oglogn),丽害))+P害))+P(害)) Observethatm., axlEslnEX./(凡)=o(~/元).Wehave ^'',札 妻^P(77"))I1z1u o.1o. C?n,f,?P(Xj"?0)?nr-1^P(IXl(凡)) 'l=J,:/n=1 C妻n,,(1ogn).P(IXI(凡))KE[IXI.(1.gIX1)"一]'=l byLerrlmas2and2.6.fHuangalldZhang(2006).Als.,n.ticethatVar(^)=.(n).By 矗=1 Lemma3.2wehaveforQ>27, i/r-2(丽z)) c 一?tr--2,JJexp{-),墨.厶?丽1凡E[Q{IXln))] +善E[IXlQ(卜1)<IXl(删善1^xo.r一 cz[IXl_1)<删而nTMi J (10.+(1O"),=l,,,一… (byLemmas2?3alld2?6ofHuangand — Zhang(2006)) ?+墨E[IXl_1)<IXl))](" Finally.bynoticingLe,nma3.3,wecompetetheproofofLemma3.4. Lemma3.5Let7?1allda>一1.Supposethattheconditions(1.3)and(1.2)issatisfied Thenforany1/2<P<pwehave I)fsut) \0<<1 +1/(1.gn)1一p I'(丽:)P(supIW(s)l.一1/(1ogn)p)+PVx>0,(3.5),0<s<1 P(INI+1/(1ogn)p)一P I)(1:I,/百)P(INI.一1/(1ogn)p)+P,Vx>0,(3.6) 第三期 whel'ePn0satisfies 张立新林正炎:在最少条件下的对数律精确渐近性317 (3.7) ProofByLennna3.1.thereexistauniversalconstantA>0andasequenceofstandard Wienelprocesses{I,T(?))suchthatfol.allQ>2, P(1naxS't计一()l>三/(1.g) (\(1ogn)p'. )Qk=l -- ,『Qcn()QE[IXIQI{IXlx/~/(1ogn))] Ontheotherhand,byLemma1.1.1ofCsbrg6andRdvdsz(1981), Let P]IIlaX.sB,d一 n)l>三V/-~n/(1ogn)) 一-PIll 删 aX ()l>三,_o1 xp {一)<Cnexp{一 一 … sup l一l n/(?.印) (1ogn)P T1leuPsatisties(3.5)and(3.6),sii1ce{(tB,)/,/;t?0)呈{T(t);t0)fol?each扎.And also, cn((1ogn)v')QE[I/(1.gn))]+Cnexp{一1n/(1.gn).) Itfollowsthat ?oc ?,t~r-2,JPKL+C?n ,l=L'l=l 一 Q/.,j(1ogn)pQE[IXIQI{IXIx/n/(1ogn)p}] KI-t"C言~,Q<)nr-l-Q/2,Q),, KI-I--C~<)] Qln+nJr-l-Q(1ogn)pQ}(1ogn)(1ogn)Q/.(1ogj)pQlQ/."+?.J.Q}'=1 usingLemmas2.3arid2.6ofHuangandZhang(2006)) c耶<,Q+n Kl+CE[IXI扑(1ogII)'p一p)Q+p+"]K<.., whenever(P一p)Q+27'P+a<一1.So,(3.7)issatisfied. ? < p p 0 <一 p 2 一 n ?? 一1一n 斗 x? ,?J? 318应用概率统计 第二十二卷 . -P??JJ)+P{?,?n>) n)_)+P{T47=-f)1?>) P2[一])+,,n Psupev~-gn一2一 )+p PIW(s)I?e2v/~gn一3)+p+ f0Iallf?(一,+),where,denedinLe蚴aLs3. 4with:/4and lisdefinedirILemnlas3.5.Also,ifnislargeenough , )P) 三啊,+) ?啊加)-P{:>) [E+)1?>) PsupIW(s)JEv/~logn+3)一一厶 … forall:?(一?+),whereedinLemma3.3with:/4.similarIy It凡1Slargeenough. P{I?Ie+3)--Pn--,n P{JsJe())P{f?Iev/~ogn—ii3)+pn+I hdsfo1'allF?(一,+). ggn=p厶+,mpIeteSthepro0fby Letllnla~S3.3,3.4all(I3. 5. ?4.ProofoftheTheorem r, Obvi … ously , , ('?''?6)all(i(1?5)爿(1.7).Itremainsforust.sh0w(1.3)(1.4)and1(. 5),an(1(1.6llO1'(1.7)(1.3).一… ?n :T R <八l ?<一 一一 mnP<一 O 曲 w 叫 , 第三期张立新林正炎:在最少条件下的对数律精确渐近性319 First,weshowthat(1.4)and(1.5)holdUlldei'(1.3).Withoutlosingofgenerality,weassume thatEX=0;-IliaEX!=1. Let>0smallenoughand{g,)besuchthat(3.1),(3.2)and(3.3) llok1.Then liI[F2一(7.一1)]n+l/2量nr一Ag:0,f\?,',l'l=1 by(3.3).Noti('ethatn(E)---+0.By(3.1).Wehavethatfol'nlargeenough, P{sl1pL0<<1 ~ /一2log—n(el(?))+卜gn P~--1-(tn?))PsupIW(s)lX/一2logn(e))-)+‰ Ontheothel'hand,1)yP1'oposition2.1 , 【-1) , {一sups)_X/一2logn(01e+.))?log)f\?''一l"=】L<s<l70J =唧r_)r(n+1/2)- (1.4)isIIO'~Vpll(】,,ed.Tilepl'oofof(1.5)issimilm.byusing(3.2)andProposition2.1 The1)I'OOfof(1.7)j(13)~llld(1.6)j(1.3)isstandardandSOomitted. References [1】Chow,Y.S.andLai,TL.,Someolle— sidedtheoremsOilthetaildistributionofsamplesumswith applicationstothelasttimeandlm'gestexcessofboundarycrossings,Trans.Amer.Math.Soc. , 208(1975),5172. Cs&.g6,M.andR,'dsz.P,Strongp7lrJ.7n0DnsinProbabilityandStatistics,Academic,NewYork, 1981. Einmahl,U.,TheDarling— ErdSstheo~'elnforSUllISofi.i.d.randomvariables,Probab.Theor'YRelat. Fields,82(1989),241—257. Feller,W.,Thelawoftheiteratedlogarithmforidenticallydistributedrandomvariables,Ann,Math., 47(1945).631—638. Gut,A.andS1)htm'11,A.,Precise~symptoticsintheBaum— KatzandDavislawoflargenumbers,J. Math.Ana1.App1.,248(2000),233—246. Huang.W.andZhaug.L.X..PreciseratesintheIaWSofthelogarithmintheHilbertspacc.,.Math. Ana1.App1.,304(2005),734—758. LaiT.L..LimittheoremsfordelayedStlnlS,Ann.Probab,,2(1975),432—440. Li;mg,H.Y.,Zhang,D.X.all(1Back,J.I.,Preciseasympototicsinthelawofthelogarithm,Manuscript, 2003. Petrov,V.V.LimitTheoremsofP~vbabilityTheory,OxfordUniversityPress,Oxford,1995. Sakhanenko,A.I..Onunimi)rovableestimatesoftherateofconvergenceintheinvarianceprinci一 1)le,InColloquiaMatl~.Soci.JdnosBolyai,32(1980),779— 783,NonparametricStatisticalInference, Buda1)est(Huugar3'). 4567890 320 [】1】 [12】 [1.3】 [14】 应用概率统计第二十二卷 Sakhanenko,A.I.,Onestimatesoftherateofconvergenceintheinvarianceprinciple,InAdvances inProbab.Theory:LimitTheoremsandRelatedProblems(A.A.Borovkov,Ed.),124—135,Springer, NewYork,1984. SakhanenkoA.I..Convergencel-ateintheinvarianceprinciplefornon—identicallydistributedvariables withexponentialliloments,InAdvancesinProbab.Theory:LimitTheoremsyorSumsoyRan dom Variables(A.A.Borovkov,Ed.),2—73,Springer,NewYork,1985. Sp~taru,A.,PreciseasymptoticsforaseriesofT.L.Lai,Proc.Amer.Math.Soc.,132(11)(200 4), 3387—3395. Zhang,L.X.,Precise1.atesillthelawoftheiteratedlogarithm,Manuscript. zju. edu.cn/zlx/mypa1)ers2/1.ateofLIL.pdf,2001 在最少条件下的对数律精确渐近性 张立新 (浙江大学系, 林正炎 杭州,310028) 设1,,…为独立同分布随机变量,记S=1+…+,=maxISkI,n1.本 \儿 文在充分必要:条件下给出了且和S的对数律之精确渐近性. 关键词:独立随机变量和的尾概率,对数律,强逼近 学科分类号:0211.4.