Tail asymptotics for extinction times of self-similar fragmentations

TL;DR

This paper characterizes the exact large-time tail behavior of extinction times in self-similar fragmentations with negative self-similarity index, revealing factors influencing this behavior and applying results to various models including random real trees.

Contribution

It provides the precise asymptotic tail distribution of extinction times, improving previous logarithmic results and analyzing the influence of fragmentation factors.

Findings

Exact large-time tail distribution derived

Asymptotic behavior of moments of the largest fragment obtained

Application to models like stable Lévy trees and beta-splitting

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.