Normalizing constants for branching processes in random environments.pdf

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1、StochasticProcessesandTheirApplications6(1978)201-211.@INorth-HollandPublishingCompanyNORMALIZINGCONSTANTSFORBRANCHINGPROCESSESINRANDOMENVIRONMENTS(B.P.R.E.)”DavidTANNYDepartmentofMathematics,UniversityofRochesttr,Rochester,NY,U.S.A.Received22May1976Revised11Way1977Normalizingconsta

2、ntsareobtainedforB.P.R.E.suchthatthelimitingrandomvariableisfinitealmosteverywhereandiszeroonlyontheextinctionsetoftheprocessw.p.1.Furthermore,thenormalizingconstantscanbechosensothattheygrowexponentialjlyfast,andsothattheratioofsuccessiveconstantsconvergesindistribution.T~I=methodo

3、fproofusedistoprovetheresultforincreasingbranchingprocesses,andthen,totransfertheresulttogeneralB.P.R.E.byemployingtherelationshipsbetweenB.P.R.E.,theassociatedB.P.R.E.,andthereducedbranchingprocess.AMS1970SubjectClassification:Primary60580;Secondary6OFl!i.branchingprocessrandomenvi

4、ronmentIII1.IndroductionLet,{Z,,}Z=Obeaone-dimensionalGalton-Watsonprocess(see[4]).Sernata171showedthatthereexistsasequenceofnormalizingconstants(c,}~=,suchthatc;Z,,convergesindistributiontoapositiverandomvariableWwhichiszeroonlyontheextinctionsetoftheprocess.In1970,Heyde,usinganexp

5、onentialmartingale,strengthenedtheresulttoalmosteverywhereconvergence.In[S],Scnazagivesashorterproofofthealmosteverywhereconvergenceand,inaddition,findsnormalizingconstantsforbranchingprocessesinavaryingenvironment(see[6]).Unfortunately,hislimitrandomvariableWmaybeinfiniteonsomeseth

6、avingpositiveprobabilityand,furthermore,maybezeroonasetstrictlylargerthantheextinctionsetoftheprocess.Inthispaper,weseektheanalogueforB.P.R.E.(see[2]forthedefinitionofB.P.R.E.)ofthenormalizationtheoremprovedforGalton-Watsonprocesse%.Inparticular,weprovethefollowingtheorem.Theorem1.L

7、et{Z,}LbeaB.P.R.E.withastatiomryanderodicenvironsequence

8、blesC,(e),depending