{"title":"Large Deviations for a Critical Galton-Watson Branching Process","authors":"Dou-dou Li, Wan-lin Shi, Mei Zhang","doi":"10.1007/s10255-024-1058-y","DOIUrl":null,"url":null,"abstract":"<p>In this paper, a critical Galton-Watson branching process {<i>Z</i><sub><i>n</i></sub>} is considered. Large deviation rates of <span>\\({S_{{Z_n}}}: = \\sum\\limits_{i = 1}^{{Z_n}} {{X_i}} \\)</span> are obtained, where {<i>X</i><sub><i>i</i></sub>, <i>i</i> ≥ 1} is a sequence of independent and identically distributed random variables and <i>X</i><sub>1</sub> is in the domain of attraction of an <i>α</i>-stable law with <i>α</i> ∈ (0, 2). One shall see that the convergence rate is determined by the tail index of <i>X</i><sub>1</sub> and the variance of <i>Z</i><sub>1</sub>. Our results can be compared with those ones of the supercritical case.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10255-024-1058-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, a critical Galton-Watson branching process {Zn} is considered. Large deviation rates of \({S_{{Z_n}}}: = \sum\limits_{i = 1}^{{Z_n}} {{X_i}} \) are obtained, where {Xi, i ≥ 1} is a sequence of independent and identically distributed random variables and X1 is in the domain of attraction of an α-stable law with α ∈ (0, 2). One shall see that the convergence rate is determined by the tail index of X1 and the variance of Z1. Our results can be compared with those ones of the supercritical case.
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