{"title":"Tail asymptotics for extinction times of self-similar fragmentations","authors":"Bénédicte Haas","doi":"10.1214/22-aihp1306","DOIUrl":null,"url":null,"abstract":"We provide the exact large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity, improving thus a previous result on the logarithmic asymptotic behavior of this tail. Two factors influence this behavior: the distribution of the largest fragment at the time of a dislocation and the index of self-similarity. As an application we obtain the asymptotic behavior of all moments of the largest fragment and compare it to the behavior of the moments of a tagged fragment, whose decrease is in general significantly slower. We illustrate our results on several examples, including fragmentations related to random real trees - for which we thus obtain the large-time behavior of the tail distribution of the height - such as the stable L\\'evy trees of Duquesne, Le Gall and Le Jan (including the Brownian tree of Aldous), the alpha-model of Ford and the beta-splitting model of Aldous.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1306","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 4
Abstract
We provide the exact large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity, improving thus a previous result on the logarithmic asymptotic behavior of this tail. Two factors influence this behavior: the distribution of the largest fragment at the time of a dislocation and the index of self-similarity. As an application we obtain the asymptotic behavior of all moments of the largest fragment and compare it to the behavior of the moments of a tagged fragment, whose decrease is in general significantly slower. We illustrate our results on several examples, including fragmentations related to random real trees - for which we thus obtain the large-time behavior of the tail distribution of the height - such as the stable L\'evy trees of Duquesne, Le Gall and Le Jan (including the Brownian tree of Aldous), the alpha-model of Ford and the beta-splitting model of Aldous.
我们提供了具有负自相似指数的自相似破碎过程消光时间尾部分布的精确大时间行为,从而改进了先前关于该尾部的对数渐近行为的结果。影响这种行为的因素有两个:位错发生时最大碎片的分布和自相似指数。作为一种应用,我们得到了最大片段的所有矩的渐近行为,并将其与标记片段的矩的行为进行比较,标记片段的矩的减少通常要慢得多。我们用几个例子来说明我们的结果,包括与随机真实树相关的碎片-因此我们获得了高度尾部分布的大时间行为-例如Duquesne, Le Gall和Le Jan的稳定L\'evy树(包括Aldous的布朗树),Ford的α模型和Aldous的β分裂模型。