Asymptotic behavior of weighted quadratic variatio

arXiv:0802.3307v1[math.PR]22Feb2008AsymptoticbehaviorofweightedquadraticvariationsoffractionalBrownianmotion:thecriticalcaseH=1/4IvanNourdin1andAnthonyR´eveillac2Abstract:WederivetheasymptoticbehaviorofweightedquadraticvariationsoffractionalBrow-nianmotionBwithHurstindexH=1/4.ThiscompletestheonlymissingcaseinaveryrecentworkbyI.Nourdin,D.NualartandC.A.Tudor.Moreover,asanapplication,wesolvearecentconjectureofK.BurdzyandJ.SwansonontheasymptoticbehavioroftheRiemannsumswithalternatingsignsassociatedtoB.Keywords:FractionalBrownianmotion;quarticprocess;changeofvariableformula;weightedquadraticvariations;Malliavincalculus;weakconvergence.Presentversion:February20081IntroductionLetBHbeafractionalBrownianmotionwithHurstindexH∈(0,1).SincetheseminalworksbyBreuerandMajor[1],DobrushinandMajor[4],GiraitisandSurgailis[5]orTaqqu[24],itiswell-knownthat•ifH∈(0,34)then1√nn−1Xk=0n2H(BH(k+1)/n−BHk/n)2−1Law−−−→n→∞N(0,C2H);(1.1)•ifH=34then1√nlognn−1Xk=0n3/2(B3/4(k+1)/n−B3/4k/n)2−1Law−−−→n→∞N(0,C234);(1.2)•ifH∈(34,1)thenn1−2Hn−1Xk=0n2H(BH(k+1)/n−BHk/n)2−1Law−−−→n→∞“Rosenblattr.v.”.(1.3)1Universit´ePierreetMarieCurieParisVI,LaboratoiredeProbabilit´esetMod`elesAl´eatoires,Boˆıtecourrier188,4placeJussieu,75252ParisCedex05,France,[e-mail]2Universit´edeLaRochelleLaboratoireMath´ematiques,ImageetApplications,AvenueMichelCr´epeau,17042LaRochelleCedex,[e-mail]1Here,CH0denotesaconstantdependingonlyonHandwhichcanbecomputedexplicitly.Moreover,theterm“Rosenblattr.v.”denotesarandomvariablewhosedistributionisthesameasthatoftheRosenblattprocessZattimeone,see(1.9)below.Now,letfbearealfunctionassumedtoberegularenough.Veryrecently,theasymptoticbehaviorofn−1Xk=0f(BHk/n)n2H(BH(k+1)/n−BHk/n)2−1(1.4)receivedalotofattention,see[6,12,13,14,16](seealsotherelatedworks[17,20,21,23]).TheinitialmotivationofsuchastudywastoderivetheexactratesofconvergenceofsomeapproximationschemesassociatedtoscalarstochasticdifferentialequationsdrivenbyBH,see[6,12,13]forprecisestatements.Butitturnedoutthatitwasalsointerestingforitselfbecauseithighlightednewphenomenawithrespectto(1.1)-(1.3).Indeed,inthestudyoftheasymptoticbehaviorof(1.4),anewcriticalvalue(H=14)hasappeared.Moreprecisely:•ifH14thenn2H−1n−1Xk=0f(BHk/n)n2H(BH(k+1)/n−BHk/n)2−1L2−−−→n→∞14Z10f′′(BHs)ds;(1.5)•if14H34then1√nn−1Xk=0f(BHk/n)n2H(BH(k+1)/n−BHk/n)2−1Law−−−→n→∞CHZ10f(BHs)dWs(1.6)forWastandardBrownianmotionindependentofBH;•ifH=34then1√nlognn−1Xk=0f(B3/4k/n)n3/2(B3/4(k+1)/n−B3/4k/n)2−1Law−−−→n→∞C3/4Z10f(B3/4s)dWs(1.7)forWastandardBrownianmotionindependentofB3/4;•ifH34thenn1−2Hn−1Xk=0f(BHk/n)n2H(BH(k+1)/n−BHk/n)2−1L2−−−→n→∞Z10f(BHs)dZs(1.8)forZtheRosenblattprocessdefinedbyZs=IX2(Ls),(1.9)whereIX2denotesthedoublestochasticintegralwithrespecttotheWienerprocessXgivenbythetransferequation(2.3)andwhere,foreverys∈[0,1],LsisthesymmetricsquareintegrablekernelgivenbyLs(y1,y2)=121[0,s]2(y1,y2)Zsy1∨y2∂KH∂u(u,y1)∂KH∂u(u,y2)du.2Evenifitisnotcompletelyobviousatfirstglance,convergences(1.1)and(1.5)wellagree.Indeed,since2H−1−12ifandonlyifH14,(1.5)isactuallyaparticularcaseof(1.1)whenf≡1.Theconvergence(1.5)isprovedin[14]whiletheothercases(1.6)-(1.8)areprovedin[16].Ontheotherhand,noticethattherelations(1.5)-(1.8)donotcoverthecriticalcaseH=14.Ourfirstmainresultcompletesthisimportant(seewhyjustbelow)missingcase:Theorem1.1.IfH=14then1√nn−1Xk=0f(B1/4k/n)√n(B1/4(k+1)/n−B1/4k/n)2−1Law−−−→n→∞C1/4Z10f(B1/4s)dWs+14Z10f′′(B1/4s)ds(1.10)forWastandardBrownianmotionindependentofB1/4andwhereC21/4=12∞Xp=−∞p|p+1|+p|p−1|−2p|p|2∞.Here,itisinterestingtocomparetheobtainedlimitin(1.10)withthoseobtainedintherecentwork[17].In[17],theauthorsalsostudiedtheasymptoticbehaviorof(1.4)butwhenthefractionalBrownianmotionBHisreplacedbyaniteratedBrownianmotionZ,thatistheprocessdefinedbyZt=X(Yt),t∈[0,1],withXandYtwoindependentstandardBrownianmotions.IteratedBrownianmotionZisself-similarofindex14andhasstationaryincrements.Thus,althoughifitisnotGaussian,Zis“close”tothefractionalBrownianmotionB1/4.ForZinsteadofB1/4,itisprovedin[17]thatthecorrectlyrenormalizedweightedquadraticvariation(whichisnoteexactlydefinedasin(1.4),butratherbymeansofarandompartitioncomposedofBrownianhittingtimes)convergesinlawtowardstheso-calledweightedBrownianmotioninrandomsceneryattimeone,definedas√2Z+∞−∞f(Xx)Lx1(Y)dWx,comparewiththeright-handsideof(1.10).Here,{Lxt(Y)}x∈R,t∈[0,1]standsforthejointlycontinuousversionofthelocaltimeprocessofY,whileWdenotesatwo-sidedstandardBrownianmotionindependentofXandY.Fornow,wetakeBH=B1/4tobeafractionalBrownianmotionwithHurstindexH=14.ThisparticularvalueofHisimportantbecausethefractionalBrownianmotionwithHurstindexH=14hasaremarkablephysicalinterpretationintermsofparticlesystems.Indeed,ifoneconsideraninfinitenumberofparticles,initiallyplacedonthereallineaccordingtoaPoissondistribution,performingindependentBrownianmotionsandundergoing“elastic”collisions,thenthetrajectoryofafixedparticle(afterrescaling)convergestoafractionalBrownianmotionwithHurstindexH=14.ThisstrikingfacthasbeenfirstpointedoutbyHarrisin[10],andth