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Behavior near the extinction time in self-similar fragmentations I: the stable case

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Date
2010
Link to item file
http://arxiv.org/abs/0805.0967
Dewey
Probabilités et mathématiques appliquées
Sujet
Probabilités
Journal issue
Annales de l'I.H.P. Probabilités et Statistiques
Volume
46
Number
2
Publication date
2010
Article pages
338-368
Publisher
Institute of Mathematical Statistics
DOI
http://dx.doi.org/10.1214/09-AIHP317
URI
https://basepub.dauphine.fr/handle/123456789/2040
Collections
  • CEREMADE : Publications
Metadata
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Author
Goldschmidt, Christina
Haas, Bénédicte
Type
Article accepté pour publication ou publié
Abstract (EN)
The stable fragmentation with index of self-similarity $\alpha \in [-1/2,0)$ is derived by looking at the masses of the subtrees formed by discarding the parts of a $(1 + \alpha)^{-1}$--stable continuum random tree below height $t$, for $t \geq 0$. We give a detailed limiting description of the distribution of such a fragmentation, $(F(t), t \geq 0)$, as it approaches its time of extinction, $\zeta$. In particular, we show that $t^{1/\alpha}F((\zeta - t)^+)$ converges in distribution as $t \to 0$ to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of $\zeta$.

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