Probability laws related to the Jacobi theta and R

ProbabilitylawsrelatedtotheJacobithetaandRiemannzetafunctions,andBrownianexcursionsPhilippeBiane,JimPitmanyandMarcYorzTechnicalReportNo.569DepartmentofStatisticsUniversityofCalifornia367EvansHall#3860Berkeley,CA94720-3860October1999.RevisedNovember22,1999AbstractThispaperreviewsknownresultswhichconnectRiemann’sintegralrepresen-tationsofhiszetafunction,involvingJacobi’sthetafunctionanditsderivatives,tosomeparticularprobabilitylawsgoverningsumsofindependentexponentialvari-ables.Theselawsarerelatedtoone-dimensionalBrownianmotionandtohigherdimensionalBesselprocesses.Wepresentsomecharacterizationsoftheseproba-bilitylaws,andsomeapproximationsofRiemann’szetafunctionwhicharerelatedtotheselaws.Keywords:Innitelydivisiblelaws,sumsofindependentexponentialvariables,Besselprocess,functionalequationAMSsubjectclassications.11M06,60J65,60E07CNRS,DMA,45rued’Ulm75005Paris,FranceyDept.Statistics,U.C.,Berkeley.ResearchsupportedinpartbyN.S.F.GrantDMS97-03961zLaboratoiredeProbabilites,UniversitePierreetMarieCurie,4PlaceJussieuF-75252ParisCedex05,France.ResearchsupportedinpartbyN.S.F.GrantDMS97-039611Contents1Introduction32Probabilisticinterpretationsofsomeclassicalanalyticformulae42.1Someclassicalanalysis............................42.2Probabilisticinterpretationof2(s).....................63Twoinnitelydivisiblefamilies73.1LaplacetransformsandLevydensities...................103.2Probabilitydensitiesandreciprocalrelations................113.3MomentsandMellintransforms.......................123.4Characterizationsofthedistributionsof2and#2............134Brownianinterpretations144.1IntroductionandNotation..........................144.2Besselprocesses................................184.3Atableofidentitiesindistribution.....................194.4SquaredBesselprocesses(Row2)......................224.5Firstpassagetimes(Row3).........................234.6Maximaandtheagreementformula(Rows4and5)............254.7Furtherentries.................................265RenormalizationoftheseriesPns.275.1Statementoftheresult............................275.2Onsumsofindependentexponentialrandomvariables..........285.3ProofofTheorem3..............................295.4ThecaseoftheL4function.........................305.5Comparisonwithothersummationmethods................306Finalremarks316.1Hurwitz’szetafunctionandDirichlet’sL-functions............316.2OtherprobabilisticaspectsofRiemann’szetafunction..........3321IntroductionInhisfundamentalpaper[63],RiemannshowedthattheRiemannzetafunction,initiallydenedbytheseries(s):=1Xn=1ns(s1)(1)admitsameromorphiccontinuationtotheentirecomplexplane,withonlyasimplepoleat1,andthatthefunction(s):=12s(s1)s=2(12s)(s)(s1)(2)istherestrictionto(s1)ofauniqueentireanalyticfunction,whichsatisesthefunctionalequation(s)=(1s)(3)forallcomplexs.Thesebasicpropertiesofandfollowfromarepresentationof2astheMellintransformofafunctioninvolvingderivativesofJacobi’sthetafunction.Thisfunctionturnsouttobethedensityofaprobabilitydistributionontherealline,whichhasdeepandintriguingconnectionswiththetheoryofBrownianmotion.Thisdistributionrstappearsintheprobabilisticliteratureinthe1950’sintheworkofFeller[24],Gnedenko[26],andTakacs[71],whoderiveditastheasymptoticdistributionasn!1oftherangeofasimpleone-dimensionalrandomwalkconditionedtoreturntoitsoriginafter2nsteps,andfoundformula(5)belowfors=1;2;.CombinedwiththeapproximationofrandomwalksbyBrownianmotion,justiedbyDonsker’stheorem[9,20,62],therandomwalkasymptoticsimplythatifY:=q2max0u1bumin0u1bu(4)where(bu;0u1)isthestandardBrownianbridgederivedbyconditioningaone-dimensionalBrownianmotion(Bu;0u1)onB0=B1=0,thenE(Ys)=2(s)(s2C):(5)whereEistheexpectationoperator.ManyotherconstructionsofrandomvariableswiththesamedistributionasYhavesincebeendiscovered,involvingfunctionalsofthepathofaBrownianmotionorBrownianbridgeinRdford=1;2;3or4.Ourmainpurposeinthispaperistoreviewthiscircleofideas,withemphasisontheprobabilisticinterpretationssuchas(4)-(5)ofvariousfunctionswhichplayanimportant3roleinanalyticnumbertheory.Forthemostpartthisisasurveyofknownresults,buttheresultofSection5maybenew.Section2reviewstheclassicalanalysisunderlying(5),andoersdierentanalyticcharacterizationsoftheprobabilitydistributionofY.Section3presentsvariousformulaerelatedtothedistributionsoftherandomvariableshand#hdenedbyh:=221Xn=1h;nn2and#h:=221Xn=1h;n(n12)2;(6)forindependentrandomvariablesh;nwiththegamma(h)densityP(h;n2dx)=dx=(h)1xh1ex(t0):(7)Ourmotivationtostudytheselawsstemsfromtheircloseconnectiontotheclassicalfunctionsofanalyticnumbertheory,andtheirrepeatedappearancesinthestudyofBrownianmotion,whichwerecallinSection4.Forexample,tomaketheconnectionwiththebeginningofthisintroduction,onehas2d=2Y2(8)whered=meansequalityindistribution.AswediscussinSection4,Brownianpathspossessanumberdistributionalsymmetries,whichexplainsomeoftheremarkablecoin-cidencesindistributionimpliedbytherepeatedappearancesofthelawsofhand#hforvarioush.Section5showshowoneoftheprobabilisticresultsofSection