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1、LocalestimationoftheHurstindexofmultifractionalBrownianmotionbyIncrementRatioStatisticmethodPierre,R.BERTRAND1,2,MehdiFHIMA2andArnaudGUILLIN21INRIASaclay2LaboratoiredeMathématiques,UMRCNRS6620&UniversitèdeClermont-FerrandII,FranceAbstractWeinvestigateheretheCent
2、ralLimitTheoremoftheIncrementRatioStatisticofamultifractionalBrownianmotion,leadingtoaCLTforthetimevaryingHurstindex.TheproofsarequitesimplerelyingonBreuer-Majortheoremsandanoriginalfreezingoftimestrategy.Asimulationstudyshowsthegoodnessoffitofthisestimator.Keywo
3、rds:IncrementRatioStatistic,fractionalBrownianmotion,localestimation,multifractionalBrownianmotion,waveletseriesrepresentation.IntroductionTheaimofthispaperisasimpleproofofCentralLimitTheorem(CLTinallthesequel)fortheconvergenceofIncrementRatioStatisticmethod(IRS
4、inallthesequel)toatimevaryingHurstindex.HurstindexisthemainparameteroffractionalBrownianmotion(fBminallthesequel),itbelongstotheinterval(0,1)anditwillbedenotebyHinallthefollowing.ForfBm,theHurstindexdrivesbothpathroughness,self-similarityandlongmemorypropertieso
5、ftheprocess.FBmwasintroducedbyKolmogorov[20]asGaussian"spirals"inHilbertspaceandthenpopularizedbyMandelbrot&VanNess[22]foritsrelevanceinmanyapplications.However,duringthetwolastdecades,newdeviceshaveallowedaccesstolargethenhugedatatsets.ThisputinlightthatfBmitse
6、lfisatheoreticalmodelandthatinreallifesituationtheHurstindexis,atleast,timevarying.Thismodel,calledmultifractionalBrownianmotion(mBm)hasbeenintroduced,independentlybyLévy-Véhel&Peltier[21]andBenassietal[9].OthergeneralizationsoffBmremainpossible,fore.g.Gaussianp
7、rocesseswithaHurstindexarXiv:1010.4849v1[math.PR]23Oct2010dependingofthescale,so-calledmultiscalefBm[5],whenHispiecewiseconstantasintheStepFractionalBrownianMotionsee[3],orawiderangeofGaussianornon-Gaussianprocessesfittedtoapplications(seeforexample[14,4]).Howeve
8、r,forsimplicityofthepresentation,inthisworkwerestrictourselvestomBm.Instatisticalapplications,weestimatethetimevaryingHurstindexthroughaCLT.Actually,CLTprovidesusconfi
