Moderate deviations for stationary sequences of bo

arXiv:0711.3924v1[math.PR]25Nov2007ModeratedeviationsforstationarysequencesofboundedrandomvariablesJ´erˆomeDedeckera,FlorenceMerlev`edeb,MagdaPeligradc1andSergeyUtevdaUniversit´eParis6,LSTA,175rueduChevaleret,75013Paris,FRANCEbUniversit´eParis6,LPMAandC.N.R.SUMR7599,175rueduChevaleret,75013Paris,FRANCEcDepartmentofMathematicalSciences,UniversityofCincinnati,POBox210025,Cincinnati,Oh45221-0025,dSchoolofMathematicalSciences,UniversityofNottingham,Nottingham,NG72RD,UKKeywords:moderatedeviation,martingaleapproximation,stationaryprocesses.MathematicalSubjectClassification(2000):60F10,60G10.AbstractInthispaperwederivethemoderatedeviationprincipleforstationarysequencesofboundedrandomvariablesundermartingale-typeconditions.Applicationstofunctionsofφ-mixingse-quences,contractingMarkovchains,expandingmapsoftheinterval,andsymmetricrandomwalksonthecirclearegiven.1IntroductionForthestationarysequence(Xi)i∈Zofcenteredrandomvariables,definethepartialsumsandthenormalizedpartialsumsprocessbySn=nXj=1XjandWn(t)=n−1/2[nt]Xi=1Xi.InthispaperweareconcernedwiththemoderatedeviationprincipleforthenormalizedpartialsumsprocessWn,consideredasanelementofD([0,1])(functionson[0,1]withleft-handlimitsandcontinuousfromtheright),equippedwiththeSkorohodtopology(seeSection14inBillingsley(1968)forthedescriptionofthetopologyonD([0,1])).Moreexactly,wesaythatthefamilyofrandomvariables{Wn,n0}satisfiestheModerateDeviationPrinciple(MDP)inD[0,1]withspeedan→0andgoodratefunctionI(.),ifthelevelsets{x,I(x)≤α}arecompactforallα∞,andforallBorelsets−inft∈Γ0I(t)≤liminfnanlogP(√anWn∈Γ)≤limsupnanlogP(√anWn∈Γ)≤−inft∈¯ΓI(t).(1)TheModerateDeviationPrincipleisanintermediatebehaviorbetweenthecentrallimitthe-orem(an=a)andLargeDeviation(an=a/n).Usually,MDPhasasimplerratefunction,1SupportedinpartbyaCharlesPhelpsTaftMemorialFundgrantandNSAgrant,H98230-07-1-0016.1inheritedfromtheapproximatedGaussianprocess,andholdsforalargerclassofdependentrandomvariablesthanthelargedeviationprinciple.DeAcostaandChen(1998)usedtherenewaltheorytoderivetheMDPforboundedfunctionalsofgeometricallyergodicstationaryMarkovchains.Puhalskii(1994)andDembo(1996)appliedthestochasticexponentialtoprovetheMDPformartingales.Startingfromthemartingalecaseandusingtheso-calledcoboundarydecompositionduetoGordin(1969)(Xk=Mk+Zk−Zk+1,whereMkisastationarymartingaledifference),Gao(1996)andDjellout(2002)obtainedtheMDPforφ-mixingsequenceswithsummablemixingrate.InthecontextofMarkovchains,thecoboundarydecompositionisknownasthePoissonequation.Startingfromthisequation,Delyon,JuditskyandLiptser(2006)provedtheMDPforn−1/2Pnk=1H(Yk),whereHisaLipschitzfunction,andYn=F(Yn−1,ξn),whereFsatisfies|F(x,z)−F(y,t)|≤κ|x−y|+L|z−t|withκ1,and(ξn)n≥1isaniidsequenceofrandomvariablesindependentofY0.Intheirpaper,therandomvariablesarenotassumedtobebounded:theauthorsonlyassumethatthereexistsapositiveδsuchthatE(eδ|ξ1|)∞.TheystronglyusedtheMarkovstructuretoderivesomeappropriatepropertiesforthecoboundary(seetheirlemma4.2).Inthispaperweproposeamodificationofthemartingaleapproximationapproachthatallowstoavoidthecoboundarydecompositionandthustoenlargetheclassofdependentsequencesknowntosatisfythemoderatedeviationprinciple.Recentornewexponentialinequalitiesareappliedtojustifythemartingaleapproximation.Theconditionsinvolvedinourresultsarewelladaptedtoalargevarietyofexamples,includingregularfunctionalsoflinearprocesses,expandingmapsoftheintervalandsymmetricrandomwalksonthecircle.Thepaperisorganizedasfollows.InSection2westatethemainresults.Adiscussionoftheconditions,clarifications,andsomesimpleexamplesandextensionsfollow.Section3describestheapplications,whileSection4isdedicatedtotheproofs.Severaltechnicallemmasareprovedintheappendix.2ResultsFromnowon,weassumethatthestationarysequence(Xi)i∈ZisgivenbyXi=X0◦Ti,whereT:Ω7→ΩisabijectivebimeasurabletransformationpreservingtheprobabilityPon(Ω,A).ForasubfieldF0satisfyingF0⊆T−1(F0),letFi=T−i(F0).BykXk∞wedenotetheL∞-norm,thatisthesmallestusuchthatP(|X|u)=0.Ourfirsttheoremanditscorollarytreattheso-calledadaptedcase,X0beingF0–measurableandsothesequence(Xi)i∈Zisadaptedtothefiltration(Fi)i∈Z.Theorem1AssumethatkX0k∞∞andthatX0isF0–measurable.Inaddition,assumethat∞Xn=1n−3/2kE(Sn|F0)k∞∞,(2)andthatthereexistsσ2≥0withlimn→∞kn−1E(S2n|F0)−σ2k∞=0.(3)Then,forallpositivesequencesanwithan→0andnan→∞,thenormalizedpartialsumsprocessesWn(.)satisfy(1)withthegoodratefunctionIσ(·)definedbyIσ(h)=12σ2Z10(h′(u))2du(4)2ifsimultaneouslyσ0,h(0)=0andhisabsolutelycontinuous,andIσ(h)=∞otherwise.ThefollowingcorollarygivessimplifiedconditionsfortheMDPprinciple,whichwillbeverifiedinseveralexampleslateron.Corollary2AssumethatkX0k∞∞andthatX0isF0–measurable.Inaddition,assumethat∞Xn=1n−1/2kE(Xn|F0)k∞∞,(5)andthatforalli,j≥1,limn→∞kE(XiXj|F−n)−E(XiXj)k∞=0.(6)ThentheconclusionofTheorem1holdswithσ2=Pk∈ZE(X0Xk).Thenexttheoremallowstodealwithnon-adaptedsequencesanditprovidesadditionalapplications.LetF−∞=Tn≥0F−nandF∞=Wk∈ZFk.Theorem3AssumethatkX0k∞∞,E(X0|F−∞)=0almostsurely,andX0isF∞-measurable.DefinetheprojectionoperatorsbyPj(X)=E(X|Fj)−E(X|Fj−1).Supposethat(6)holds