Spectral Theory of Branching Processes I The Case of Discrete Spectrum (1).pdf

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1、Z.W~hrscheinlichkeitstheorieverw.Geb.5,6--33(1966)SpectralTheoryofBranchingProcesses.I*TheCaseofDiscreteSpectrumSAMtT]~LKA~Lr~andJAMESMcGrEGORReceivedSeptember28,1965w1.IntroductionAone-dimensionalMarkerbranchingprocessmaybecharacterizedasfollows.Anorganism,attheendofitslifetime(offixedduration)

2、,producesarandomnumber~ofoffspringwithprobabilitydistribution(1)Pr{~=k}=a~~=O,1,2....whereasusualak~0~ak:1.k=0Alloffspringactindependentlywiththesamelifetimeanddistributionofprogeny.ThepopulationsizeX(n)atthenthgenerationisatemporallyhomo-geneousMarkovchainwhosetransitionprobabilitymatrixis(2)P~

3、i:Pr{X(n+1)=jIX(n):i}:Pr{$1q-~2+"'"+~=j}where~'sareindependentobservationsofarandomvariablewiththeprobabilitylaw(I).Anequivalentwaytoexpress(2)isthroughitsgeneratingfunctionwhichissimply(3)~Pi1sl=[/(s)]/i=0,1....j~0whereook=oItisafamiliarfactthatthen-steptransitionprobabilitymatrixP!?=pr{X(n)=i[

4、x(0)=i}possessesthegeneratingfunction(4)~p~;o#=[In(s)]r1=0where(5)/,~(~)=/n-~(/(~))isthenthfunctionaliterateof/(s).*ResearchsupportedinpartbyContractsONR225(28)andNIHUSFHS10452atStanfordUniversity.SpectralTheoryofBranchingProcesses.I7AfinitestateMarkovtransitionmatrixPijordinarilyadmitsaspectral

5、re-presentationoftheformp.n.__(6)v--~~rnYJt(r)0j(r)rwhere~r(r=1,2....)aretheeigenvaluesofthematrixPii,yJi(r)denotestheithcomponentoftherthrighteigenvectorand0j(r)denotesthejthcomponentoftherthlefteigenvector.Thesystem{~0~(r),0~(r)}~'__1ischosentobebiortho-normal.Therepresentation(6)iscertainlyva

6、lidwhenalleigenvaluesaresimple,i.e.,noelementarydivisorsarise.Forinfinitetransitionprobabilitymatricesthepos-sibilityofaspectraldecompositionlike(6)israre.Theexistenceofeigenvaluesisnotevenassuredandindeed,continuousspectrumisusuallypresent.InthecasewhereP~Sisthetransitionmatrixofaone-dimensiona

7、lrandomwalk(see[7])thenPisaJacobimatrixandageneralizedspectralrepresentationexists.InthiscaseametriccanbeintroducedsuchthatPbecomesself-adjointandtheclassicalspectralresolutionofHilbertspacetheoryisavailable.Thedeviceofsymme