{"title":"Controlled Galton-Watson process and its asymptotic behavior","authors":"T. Fujimagari","doi":"10.2996/KMJ/1138847159","DOIUrl":null,"url":null,"abstract":"1. In a stochastic population process described as a Galton-Watson process each individual splits independently according to a given probability law and new born particles constitute the following generation. In addition to the independence in splitting the law of splitting of each individual depends on neither the generation to which an individual belongs nor the existence of the other individuals of the same generation and is common to all individuals. We shall consider a somewhat generalized Galton-Watson process in the sense that the law of splitting of each individual depends on the total number of individuals of the same generation and the other independence properties are reserved. The object of this note is to study asymptotic behaviors of such processes, although we can hardly obtain any complete results up to now except some partial results. The difficulties in analysing the process will be due to the dependence introduced above from which it no longer holds such as the iteration property of a generating function which plays a fundamental role in GaltonWatson processes. We shall formulate the process under consideration as follows. Let Zn be the size or the total number of individuals which belong to the n-th generation and given a sequence of probability distributions £>(i)= \\pr(i): rΞ>0}, z=0, 1,2, ••• oo","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"10 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"29","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kodai Mathematical Seminar Reports","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2996/KMJ/1138847159","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 29
Abstract
1. In a stochastic population process described as a Galton-Watson process each individual splits independently according to a given probability law and new born particles constitute the following generation. In addition to the independence in splitting the law of splitting of each individual depends on neither the generation to which an individual belongs nor the existence of the other individuals of the same generation and is common to all individuals. We shall consider a somewhat generalized Galton-Watson process in the sense that the law of splitting of each individual depends on the total number of individuals of the same generation and the other independence properties are reserved. The object of this note is to study asymptotic behaviors of such processes, although we can hardly obtain any complete results up to now except some partial results. The difficulties in analysing the process will be due to the dependence introduced above from which it no longer holds such as the iteration property of a generating function which plays a fundamental role in GaltonWatson processes. We shall formulate the process under consideration as follows. Let Zn be the size or the total number of individuals which belong to the n-th generation and given a sequence of probability distributions £>(i)= \pr(i): rΞ>0}, z=0, 1,2, ••• oo