On the distribution of ranked heights of excursion

TheAnnalsofProbability2001,Vol.29,No.1,361–384ONTHEDISTRIBUTIONOFRANKEDHEIGHTSOFEXCURSIONSOFABROWNIANBRIDGE1ByJimPitmanandMarcYorUniversityofCalifornia,BerkeleyandUniversit´ePierreetMarieCurieThedistributionofthesequenceofrankedmaximumandminimumvaluesattainedduringexcursionsofastandardBrownianbridge
Bbrt0≤t≤1isdescribed.TheheightMbr+jofthejthhighestmaximumoverapositiveexcursionofthebridgehasthesamedistributionasMbr+1/j,wherethedistributionofMbr+1=sup0≤t≤1BbrtisgivenbyL´evy’sfor-mulaP
Mbr+1x=e−2x2.TheprobabilitydensityoftheheightMbrjofthejthhighestmaximumofexcursionsofthereflectingBrownianbridge
Bbrt0≤t≤1isgivenbyamodificationoftheknownθ-functionseriesforthedensityofMbr1=sup0≤t≤1Bbrt.Theseresultsareobtainedfromamoregeneraldescriptionofthedistributionofrankedvaluesofahomoge-neousfunctionalofexcursionsofthestandardizedbridgeofaself-similarrecurrentMarkovprocess.Contents1.Introduction2.Bridgesandexcursionsofaself-similarMarkovprocess3.ProofsofTheorems1and24.ApplicationstoBesselprocesses5.Evaluationsattimeτ16.Dualformulas7.Thejointlawofτ1andFj
τ11.Introduction.LetBbr=
Bbrt0≤t≤1beastandardBrownianbridge,thatis,
Bbrt0≤t≤1d=
Bt0≤t≤1B1=0where
Btt≥0isastandardone-dimensionalBrownianmotion.See[35]forbackground.TherandomopensubsettBbrt=0of[0,1]isacountableunionofmaximaldisjointintervals(ab),calledexcursionintervalsofBbr,suchthatBbra=Bbrb=0andeitherBbrt0forallt∈
ab(apositiveexcursioninterval)orBbrt0forallt∈
ab(anegativeexcursioninterval).LetMbr+1≥Mbr+2≥···0ReceivedSeptember1999;revisedJune2000.1SupportedinpartbyNSFGrantDMS-97-03961AMS2000subjectclassification.60J65.Keywordsandphrases.Brownianbridge,Brownianexcursion,Brownianscaling,localtime,self-similarrecurrentMarkovprocess,Besselprocess.361362J.PITMANANDM.YORbetherankeddecreasingsequenceofvaluessupt∈
abBbrtobtainedas(ab)rangesoverallpositiveexcursionintervalsofBbr.Similarly,letMbr−1≥Mbr−2≥···0therankedvaluesof−inft∈
abBbrtas(ab)rangesoverallnegativeexcursionintervalsofBbrandletMbr1≥Mbr2≥···0betherankedvaluesofsupt∈
abBbrtas(ab)rangesoverallexcursioninter-valsofBbr.Onemotivationforstudyofthesequence
Mbrjisthatthissequencedescribestheasymptoticdistributionasn→∞oftherankedheightsoftreecomponentsoftherandomdigraphgeneratedbyauniformlydistributedrandommappingofann-elementsettoitself[1].NotethatMbr+1=sup0≤t≤1BbrtMbr−1=−inf0≤t≤1BbrtMbr1=sup0≤t≤1Bbrt=Mbr+1∨Mbr−1(1)Themainpurposeofthispaperistodescribeasexplicitlyaspossiblethelawsofthedecreasingrandomsequencesintroducedabove.Inparticular,weobtaintheresultsstatedinthefollowingtwotheorems.Someoftheresultsofthispaperwerepresentedwithoutproofin[32].Theorem1.Foreachj=12thecommondistributionofMbr+jandMbr−jisdeterminedbytheformulaP
Mbr+jx=e−2j2x2x≥0(2)whilethatofMbrjisdeterminedbyP
Mbrjx=2j∞
n=0−jne−2
n+j2x2x≥0(3)Formula(2)amountstotheidentitiesindistributionMbr+jd=Mbr+1jd=1jε2(4)forj=12whereεdenotesastandardexponentialvariable.Thesec-ondidentityin(4)isL´evy’s[25]well-knowndescriptionofthedistributionofsup0≤t≤1Bbrt.Despiteitssimplicity,thefirstidentityin(4)doesnotseemobvi-ouswithoutcalculation.Thecasej=1of(3)isthewell-knownKolmogorov–SmirnovformulaforthedistributionofMbr1=sup0≤t≤1Bbrt,whicharisesintheasymptotictheoryofempiricaldistributionfunctions[40,12,39,26]:P
Mbr1x=2∞
k=1
−1k−1e−2k2x2=1−θ3π22πix2(5)RANKEDHEIGHTSOFBROWNIANEXCURSIONS363whereθ3
zt=∞
n=−∞eiπn2tcos
2nzistheclassicalJacobithetafunctiondefinedfort=
t+it∈witht0.Formula(3)showsthereisnorelationassimpleas(4)betweenthedistributionofMbrjforj1andthatofMbr1.DefinetheintensitymeasureνMforthesequence
MbrjbyνM
A=E∞
j=11
Mbrj∈AforBorelsubsetsAof
0∞,anddefineνM+similarlyintermsof
Mbr+j.Formula(2)impliesthattheseintensitymeasuresνMandνM+aregivenbytheformulaνM
x∞=2νM+
x∞=2∞
j=1e−2j2x2=θ302πix2−1=θ2πx2−1(6)wherefort0,θ
t=∞
n=−∞e−πn2t=θ3
0it(7)Notethestrikingparallelbetween(5)and(6).Wenowexplainhowformula(6)isrelatedtotheformulaofChung[7]forthedistributionofthemaximumM∗ofastandardBrownianexcursion,thatis,P
M∗≤x=θ2πx2+4πx2θ2πx2x0(8)whereθisthederivativeofθ.Riemann[36]gavetheformula12s
s−1∞0t
s/2−1
θ
t−1dt=2ξ
s=s
s−1π−s/2
s/2∞
n=11ns
s1(9)anddeducedfromitandtheclassicalfunctionequation,θ
t=t−1/2θ
t−1t0that(9)definesauniqueentirefunctionξwhichsatisfiesthefunctionalequationξ
s=ξ
1−ss∈364J.PITMANANDM.YORAsshownbyBianeandYor[4],Chung’sformula(8)forP
M∗≤xisequiva-lenttothefollowingexpressionoftheMellintransformofM∗:EMs∗=π2s/22ξ
ss∈(10)Seealso[45,3]forreviewsofthiscircleofideasandotherinterpretationsofθ
tinthecontextofBrownianmotion.ThesedescriptionsofthedistributionofM∗arerelatedtoourdescription(6)oftheintensitymeasureνMforthesequence
Mbrjviatheknownresult[42,30]thattheintensitymeasureforthelengthsofexcursionsofthebridgeνV
A=E∞
j=11
Vbrj∈AwhereVbrjisthelengthofthejthlongestintervalcomponentoftherandomsubsettBbrt=0of[0,1],isdeterminedbythedensityνV
dvdv=12v3/20v1(11)Indeed,byconditioningonthelengthsofalltheexcursionsoftheBrownianbridge