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1、ABranchingProcessAllowingImmigrationAuthor(s):C.R.HeathcoteSource:JournaloftheRoyalStatisticalSociety.SeriesB(Methodological),Vol.27,No.1(1965),pp.138-143Publishedby:WileyfortheRoyalStatisticalSocietyStableURL:http://www.jstor.org/stable/2984491.Accessed
2、:17/11/201308:32YouruseoftheJSTORarchiveindicatesyouracceptanceoftheTerms&ConditionsofUse,availableat.http://www.jstor.org/page/info/about/policies/terms.jsp.JSTORisanot-for-profitservicethathelpsscholars,researchers,andstudentsdiscover,use,andbuildupona
3、widerangeofcontentinatrusteddigitalarchive.Weuseinformationtechnologyandtoolstoincreaseproductivityandfacilitatenewformsofscholarship.FormoreinformationaboutJSTOR,pleasecontactsupport@jstor.org..WileyandRoyalStatisticalSocietyarecollaboratingwithJSTORtod
4、igitize,preserveandextendaccesstoJournaloftheRoyalStatisticalSociety.SeriesB(Methodological).http://www.jstor.orgThiscontentdownloadedfrom58.60.63.199onSun,17Nov201308:32:50AMAllusesubjecttoJSTORTermsandConditions138[No.1,ABranchingProcessAllowingImmigra
5、tionByC.R.HEATHCOTEAustralianNationalUniversity[ReceivedSeptember1964]SUMMARYItiswellknownthatintheabsenceofimmigration,apopulationoflikeparticlesdevelopingundertheusuallawsforbranchingprocesseseitherincreasesunboundedlywithtimeorbecomesextinct(Feller,19
6、57,Chapter12;Harris,1963,Chapter1).Ifmigrationintothesystemispermitted,thenitisclearthatundercertainconditionsaproperstationarydistributionforpopulationsizewillexist.ThatthisisthecasehasbeenshownbyBartlett(1956,Section3.41)forprocessesincontinuoustime,an
7、dthepresentnoteisconcernedwiththesameproblembutconsideringdiscretegenerations.OurresultforthegeneratingfunctionofthestationarydistributionofpopulationsizereducestoaformanalogoustoBartlett'sresultwhentheimmigrationdistributionisPoisson.1.INTRODUCTION:THEM
8、ODELSUPPOSEwehaveasingleancestorformingthe0thgenerationwhichhasprobabilityaj(j=0,1,...)ofproducingjoffspring;wewrite00A(x)=Exiaj,Hta=Ejaj.0jIfY.denotesthenumberofnthgenerationdescendantsfromthisindividual,then00An(x)=ixjP(
