Revisiting offspring maxima in branching processes

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1、RevisitingOffspringMaximainBranchingProcessesGeorgeP.YanevDepartmentofMathematicsandStatisticsUniversityofSouthFloridaTampa,Florida33620e-mail:gyanev@cas.usf.eduAbstractWepresentaprogressreportforstudiesonmaximarelatedtooffspringinbranchingprocesses.Wesummarizeandd

2、iscussthefindingsonthesubjectthatappearedinthelasttenyears.Someoftheresultsarerefinedandillustratedwithnewexamples.1IntroductionThereisasignificantamountofresearchinthetheoryofbranchingprocessesdevotedtoextremevalueproblemsconcerningdifferentpopulationcharacteristics

3、.Thehistoryofsuchstudiesgoesbacktotheworksin50-iesbyZolotarev[26]andUrbanik[23](seealso[6])whoconsideredthemaximumgenerationsize.Ourgoalhereistosummarizeanddiscussresultsonmaximarelatedtotheoffspring.Papersdirectlyaddressingthisareaofstudyhavebeguntoappearinthelas

4、ttenyears(thoughsee”heromothers”examplein[7].)LetMndenotethemaximumoffspringsizeofallindividualslivinginthe(n−1)-stgenerationofabranchingprocess.Thisisamaximumofrandomnumberofindependentandidenticallydistributed(i.i.d.)integer-valuedrandomvariables,wheretherandomi

5、ndexisthepopulationsizeoftheprocess.Mnhastwocharacteristicfeatures:(i)thei.i.d.randomvariablesareinteger-valuedand(ii)thedistributionoftherandomindexarXiv:math/0610647v1[math.PR]21Oct2006isconnectedtothedistributionofthetermsinvolvedthroughthebranchingmechanism.T

6、hesetwocharacteristicsdistinguishthesubjectmattermaximaamongthosestudiedinthegeneralextremevaluetheory.Thestudyofthesequence{Mn}mightbemotivatedindifferentways.Itprovidesafertilitymeasurecharacterizingthemostprolificindividualinonegeneration.Italsomeasuresthemaximu

7、mlitter(orfamily)size.Inthebranchingtreecontext,itisthemaximumdegreeofavertex.TheasymptoticbehaviorofMngivesussomeinformationabouttheinfluenceofthelargestfamiliesonthesizeandsurvivaloftheentirepopulation.1Thepaperisorganizedasfollows.Nextsectiondealswithmaximainsi

8、mplebranch-ingprocesseswithorwithoutimmigration.InSection3wederiveresultsaboutmaximaofatriangulararrayofzero-inflatedgeometricvariables.Laterweapplythesetobranc