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1、Z.Wahrseheinlichkeitstheorieverw.Geb.5,34--54(1966)SpectralRepresentationofBranchingProcesses.II*CaseofContinuousSpectrumSAMV~LKA~Ln~andJAMESMcG~ooRReceivedSeptember28,1965w1.IntroductionThisdiscussionisanaturalsequeltotheprecedingpaper.Wereferthereadertotheintroductionof[3]w
2、hichcontainsbackgroundmaterialfortheproblemunderconsiderationhere.Considerabranchingprocessinducedbyaprobabilitygeneratingfunctionco/(s)~~ak8~,ak~0(/c~0,1,...)and/(1)----1.Weassumethroughoutthatk=0/(0)>0.ThebranchingprocessX(t),t=0,1,2....isaMarkerchainonthenon-negativeintege
3、rswhosetransitionprobabilitiesP~j-~P{X(t-~1)=i]X(t)----i}aredefinedbyco(1)~Pi~s~=[/(s)]i,i=0,1,2....j=0OuraimistoinvestigatethespectralpropertiesofthematrixP=[]PiJ[li.j=0.If]n(s)~]n-1(/(s))denotetheiteratesof](8)thenitisquitefamiliarthatthensteptransitionprobabilitymatrixposs
4、essesthegeneratingfunction(2)~P58J=[/.(8)]~.j=OIn[3]wetreatedthecasewherem=]'(1)>1orm~1.Inthecasem<1werequiredthat](s)beanalyticat1.ItturnedoutundermildregularityconditionsthatPanditsiteratesadmitaspectraldecompositionoftheformoo(3)P~~-~cnr0I(r)y~l(r)i,~~-O,1,2....r=0n=0,1,2.
5、...wherec~['(q)andqisthesmallestpositivesolutionof[(s)=s.ByintroducinganappropriateHilbertspacewealsoprovedthatPactsasacompletelycontinuoustransformation.Whenm=/'(1)=1,PceasestohaveeigenvectorsandinfactPhasonly"continuousspectrum"Throughout,theremainderofthispaperitisassumedt
6、hatm=]'(1)=1and](8)isanalyticinacircleofradius1+s,e>0.*ResearchsupportedinpartbyContractsONR225(28)andNIttUSPHS10452atStanfordUniversity.SpectralRepresentationofBranchingProcesses.II35Theprimeobjectiveofthispaperistoestablish,forthecasem~/'(1)=1,aspectralrepresentationofthefo
7、rmco(4)p!~)~_Se_n~Qj(~)dyj~(~),i,j=0,1,2....0n=1,2,...where~p~isofboundedvariationon[0,oo)andQj(~)isapolynomialinthevariableofdegree?',whichvanishesat~~0ff?"~1.Wewillprovethevalidityof(4)subjecttothefollowingcondition.ConditionI.Letg(s)beaprobabilitygeneratingfunctionregulara
8、ts~1.Ifg(s)isnotaconstanttheng'(1)>0andtheinversefunctiong-1(s)isthe
