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Pré-Publication, Document De Travail Année : 2022

Tail asymptotics for extinction times of self-similar fragmentations

Résumé

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.

Dates et versions

hal-03858844 , version 1 (17-11-2022)

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Bénédicte Haas. Tail asymptotics for extinction times of self-similar fragmentations. 2022. ⟨hal-03858844⟩
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