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1、StochasticProcessesandtheirApplications48(1993)237-249237North-HollandThesupercriticalGalton-Watsonprocessinvaryingenvironments-Seneta-HeydenormingJ.D.BigginsDepartmentofProbabilityandStatistics,TheUniversityofShefield,UKJ.C.DSouza*DepartmentofActuarialMathematicsandStatistics,Heri
2、ot-WattUniversity,Edinburgh,UKReceived10December1991Revised5June1992AnaturalsufficientconditionisgivenforaGalton-Watsonprocessinavaryingenvironmenttohaveasinglerateofgrowththatobtainsthroughoutthesurvivalsetoftheprocess.InthehomogeneousprocessthegrowthrateisprovidedbytheusualSeneta
3、-Heydenorming.branchingprocesses*varyingenvironments1.IntroductionWeconsidertheGalton-Watsonprocessinvaryingenvironments(Z,,}definedby&=l,z,+,=where{X,,j;i}areindependentidenticallydistributedcopiesofarandomvariableX,,.Welettheoffspringmeaninthenthgenerationbep,,;thusp,,=EX,,.Thenn
4、-lE(Z,IZ,=l)=npiforn>kgO,i=kand,inparticular,lettingm,=EZ,,weseethatm,=n:zdpi.Theaimofthispaperistogiveasufficientconditionfor(Z,}tohaveexactlyonerateofgrowth,inthesensethatCorrespondenceto:Dr.J.D.Biggins,DepartmentofProbabilityandStatistics,TheUniversityofSheffield,P.O.Box597,Shef
5、fieldSIO2UN,UK.*SupportedbyanSERCstudentship.0304.4149/93/$6,0001993-ElsevierSciencePublishersB.V.Allrightsreserved238J.D.Biggins,J.C.DSouza/Seneta-Heydenorminginvaryingenvironmentsthereisasequenceofconstants{C,)suchthatZ,/C,,convergestoafinitelimitthatisstrictlypositiveonthesurviv
6、alset,{Z,-+CO}.Itisworthnotingthatthisisonerateofgrowthinthestrongestsense.AweakersensewouldbeforthelimitofZ,,/C,,tobestrictlypositiveononlypartofthesurvivalset,butwithnoothernormalizationgivingafinitepositivelimitonanypartoftheremainder.TheexampleinMacPheeandSchuh(1983)showsthatit
7、is,ingeneral,possiblefor{Z,}toexhibitmorethanonerateofgrowth.Wewillneedatailconditionontheoffspringdistributionsandagrowthconditionontheproductoftheoffspringmeans.Thesearesuppliedthroughthefollowingdefinitions.WeshallsaythatarandomvariableXdominates{X,/E_L~}ifP(X>x)ZP(X,/P~>x)foral
8、lx,andthattheprocessisunif
