Asymptotics for the small fragments of the fragmentation at nodes

Research paper by Romain Abraham, Jean-François Delmas

Indexed on: 08 Mar '06Published on: 08 Mar '06Published in: Mathematics - Probability


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.