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时间:2019-08-01
《Revisiting offspring maxima in branching processes》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
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
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