# Asymptotics for the small fragments of the fragmentation at nodes

Abstract : We consider the fragmentation at nodes of the Lévy continuous random tree introduced in a previous paper. In this framework we compute the asymptotic for the number of small fragments at time $\theta$. This limit is increasing in $\theta$ and discontinuous. In the $\alpha$-stable case the fragmentation is self-similar with index $1/\alpha$, with $\alpha \in (1,2)$ and the results are close to those Bertoin obtained for general self-similar fragmentations but with an additional assumtion which is not fulfilled here.
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https://hal.archives-ouvertes.fr/hal-00020262
Contributor : Jean-François Delmas <>
Submitted on : Wednesday, March 8, 2006 - 4:05:46 PM
Last modification on : Thursday, May 3, 2018 - 3:32:06 PM
Document(s) archivé(s) le : Saturday, April 3, 2010 - 9:11:47 PM

### Citation

Romain Abraham, Jean-François Delmas. Asymptotics for the small fragments of the fragmentation at nodes. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2007, 13 (1), pp.211-228. ⟨10.3150/07-BEJ6045⟩. ⟨hal-00020262⟩

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