Central limit theorem for stationary linear proces

arXiv:math/0509682v2[math.PR]25Sep2006TheAnnalsofProbability2006,Vol.34,No.4,1608–1622DOI:10.1214/009117906000000179cInstituteofMathematicalStatistics,2006CENTRALLIMITTHEOREMFORSTATIONARYLINEARPROCESSESByMagdaPeligrad1andSergeyUtevUniversityofCincinnatiandUniversityofNottinghamWeestablishthecentrallimittheoremforlinearprocesseswithdependentinnovationsincludingmartingalesandmixingaletypeofassumptionsasdefinedinMcLeish[Ann.Probab.5(1977)616–621]andmotivatedbyGordin[SovietMath.Dokl.10(1969)1174–1176].Indoingsoweshallpreservethegeneralityofthecoefficients,includ-ingthelongrangedependencecase,andweshallexpressthevarianceofpartialsumsinaformeasytoapply.Ergodicityisnotrequired.1.Introduction.Let(ξi)i∈Zbeastationarysequenceofrandomvari-ableswithE[ξ20]∞andE[ξ0]=0.Let(ai)i∈Zbeasequenceofrealnum-berssuchthatPi∈Za2i=A∞anddenotebyXk=∞Xj=−∞ak+jξj,Sn=nXk=1Xk,(1)bn,j=aj+1+···+aj+nandb2n=∞Xj=−∞b2n,j.Theso-callednoncausallinearprocess(Xk)k∈Ziswidelyusedinavarietyofappliedfields.Itisproperlydefinedforanysquaresummablesequence(ai)i∈Zifandonlyifthestationarysequenceofinnovations(ξi)i∈Zhasaboundedspectraldensity.Ingeneral,thecovariancesof(Xk)k∈Zmightnotbesummablesothatthelinearprocessmightexhibitlongrangedepen-dence.AnimportantquestionistodescribetheasymptoticpropertiesofthevarianceandtheasymptoticbehaviorofSnproperlynormalized.InthisReceivedMay2005;revisedSeptember2005.1SupportedinpartbyaCharlesPhelpsTaftMemorialFundGrantandNSAGrantH98230-05-01-0066.AMS2000subjectclassifications.60F05,60G10,60G42,60G48.Keywordsandphrases.Ergodictheorem,centrallimittheorem,stationarylinearpro-cess,martingale.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofProbability,2006,Vol.34,No.4,1608–1622.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12M.PELIGRADANDS.UTEVpaperweshalladdressboththesequestions.AsimpleresultwithveryusefulconsequencesiscontainedinLemmaA.3(iii).Itturnsoutthat,whentheinnovationshaveacontinuousspectraldensityf(x),thevarianceofSnisasymptoticallyproportionaltof(0)b2n,uptoanumericalconstant.ThisfactsuggeststofurtherstudytheasymptoticdistributionofSn/bn.Asweshallseeinthispaper,ifthesequence(ξi)i∈Zisamartingaledifferencesequenceoritspartialsumscanbeapproximatedinacertainwaybymartingales,then,despitethelongrangedependence,Sn/bnsatisfiesacertaincentrallimittheorem.Toallowforflexibilityinapplications,wedefineastationaryfiltrationasin[17].Weassumethatξi=g(Yj,j≤i),where(Yi)i∈Zisanunderlyingstationarysequence.DenotebyIitsinvariantsigmafieldandby(Fi)i∈ZanincreasingfiltrationofsigmafieldsFi=σ(Yj,j≤i).Thepair[(Fi)i∈Z;I]willbecalledastationaryfiltration.Forthecasewhenforeveryi,ξi=Yj,andg(Yj,j≤i)=Yi,thenFiissimplythesigmaalgebrageneratedbyξj,j≤i.Inthesequelk·k2denotesthenorminL2,kXk2=(E[X]2)1/2.Weshallestablishthefollowingresult:Theorem1.Let(ξi)i∈ZbeastationarysequencewithE[ξ21]∞,E[ξ0]=0andstationaryfiltration[(Fi)i∈Z;I].Define(Xk)k≥1,Snandbnasaboveandassumebn→∞asn→∞.AssumethatΓj=∞Xk=0|E[ξkE(ξ0|F−j)]|∞foralljand(2)1ppXj=1Γj→0asp→∞.Then,(ξi)i∈Zhasacontinuousspectraldensityf(x)andthereisanonnega-tiverandomvariableηmeasurablewithrespecttoIsuchthatn−1E((Pnk=0ξk)2|F0)→ηinL1asn→∞andE(η)=2πf(0).Inaddition,limn→∞Var(Sn)b2n=2πf(0)and(3)Snbn=⇒√ηNindistributionasn→∞,whereNisastandardnormalvariableindependentofη.Moreover,ifthesequence(ξi)i∈Zisergodicandcondition(2)issatisfied,thenthecentrallimittheoremin(3)holdswithη=2πf(0).ThefollowingcorollaryextendstheprojectiveCLTtheoremofVolny[22](which,inturn,wasinspiredbyHeyde[11],Theorem2)andCorol-lary2(mixingaletypeCLT)ofMaxwellandWoodroofe[17]todependentCLTFORSTATIONARYLINEARPROCESSES3sequencesgeneratedbylinearprocessesand,inaddition,provestheconti-nuityofthecorrespondingspectraldensity.ThiscorollaryalsodevelopsaresultbyWuandMin[24]whoconsideredthecaseofabsolutesummableweights.Corollary2.Let(ξi)i∈ZbeastationarysequencewithE(ξ21)∞,E[ξ0]=0andstationaryfiltration[(Fi)i∈Z;I].Considertheprojectionop-eratorPi(Y)=E[Y|Fi]−E[Y|Fi−1]andassumethatE(ξ0|F−∞)=0almostsurelyand∞Xi=1kP−i(ξ0)k2∞.(4)Then,theconclusionofTheorem1holds.Inparticular,(4)issatisfiedif∞Xn=1n−1/2kE(ξn|F0)k2∞.(5)Tocommentontheconditionsusedinourresults,firstwementionthatassumption(2)impliesthattheinitialsequence(ξi)i∈ZsatisfiestheGordinmartingaleapproximationcondition(8)definedlater.Variousconditionsareknowntobesufficientfor(8),suchastheoriginalGordincondition,supnkE(ξ1+···+ξn|F0)k2∞anditsmodificationsintroducedin[11],Theorem1,orin[9],Theorem5.2,in[5,7,17,19].Byconsideringtelescop-ingsumsξn=Qn−Qn−1withthestationarysequence(Qi)i∈Zhavinganunboundedspectraldensity,onecaneasilyshowthatthoseconditionsarenotenoughfor(3).Ontheotherhand,examplessimilartothosein[22],Theorem7,showthattheGordintypeconditionsmentionedabove,imposedtopartialsums,arenotnecessaryfor(3)and(4).Asamatteroffact,weshallconstructanexampletoshowthattheconditionsofCorollary2areoptimal.Proposition3.Letψibeasequenceofnonnegativenumberssuchthatψn→0asn→∞.Then,thereexistsastrictlystationaryergodicsequence(ξi)i∈Zwithunboundedspectraldensitysuchthat