Asymptotics for the small fragments of the fragmentation at nodes

Details Asymptotics for the small fragments of the fragmentation at nodes Asymptotics for the small fragments of the fragmentation at nodes

2006-03-08

arxiv.org/abs/math/0603192v1

Bernoulli 13 (01/2007) 211-228

Mathematics

Probability

Scientific paper

10.3150/07-BEJ6045

We consider the fragmentation at nodes of the L\'{e}vy 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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