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

Christina Goldschmidt; Bénédicte Haas

Annales de l'I.H.P. Probabilités et statistiques (2010)

- Volume: 46, Issue: 2, page 338-368
- ISSN: 0246-0203

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topGoldschmidt, Christina, and Haas, Bénédicte. "Behavior near the extinction time in self-similar fragmentations I : the stable case." Annales de l'I.H.P. Probabilités et statistiques 46.2 (2010): 338-368. <http://eudml.org/doc/239331>.

@article{Goldschmidt2010,

abstract = {The stable fragmentation with index of self-similarity α∈[−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1+α)−1–stable continuum random tree below height t, for t≥0. We give a detailed limiting description of the distribution of such a fragmentation, (F(t), t≥0), as it approaches its time of extinction, ζ. In particular, we show that t1/αF((ζ−t)+) converges in distribution as t→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 ζ. In addition, we prove that the logarithms of the sizes of the largest fragment and last fragment to disappear, at time (ζ−t)+, rescaled by log(t), converge almost surely to the constant −1/α as t→0.},

author = {Goldschmidt, Christina, Haas, Bénédicte},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {stable Lévy processes; height processes; self-similar fragmentations; extinction time; scaling limits},

language = {eng},

number = {2},

pages = {338-368},

publisher = {Gauthier-Villars},

title = {Behavior near the extinction time in self-similar fragmentations I : the stable case},

url = {http://eudml.org/doc/239331},

volume = {46},

year = {2010},

}

TY - JOUR

AU - Goldschmidt, Christina

AU - Haas, Bénédicte

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

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2010

PB - Gauthier-Villars

VL - 46

IS - 2

SP - 338

EP - 368

AB - The stable fragmentation with index of self-similarity α∈[−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1+α)−1–stable continuum random tree below height t, for t≥0. We give a detailed limiting description of the distribution of such a fragmentation, (F(t), t≥0), as it approaches its time of extinction, ζ. In particular, we show that t1/αF((ζ−t)+) converges in distribution as t→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 ζ. In addition, we prove that the logarithms of the sizes of the largest fragment and last fragment to disappear, at time (ζ−t)+, rescaled by log(t), converge almost surely to the constant −1/α as t→0.

LA - eng

KW - stable Lévy processes; height processes; self-similar fragmentations; extinction time; scaling limits

UR - http://eudml.org/doc/239331

ER -

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