Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Asymptotic laws for nonconservative self-similar fragmentations

Auteur(s):

Code(s) de Classification MSC:

• 60F25 $L^p$-limit theorems
• 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Résumé: We consider a self-similar fragmentation process in which the generic particle of mass $x$ is replaced by the offspring particles at probability rate $x^\alpha$, with positive parameter $\alpha$. The total of offspring masses may be both larger or smaller than $x$ with positive probability. We show that under certain conditions the typical mass in the ensemble is of the order $t^{-1/\alpha}$ and that the empirical distribution of masses converges to a random limit which we characterise in terms of the reproduction law.

Mots Clés: Strong asymptotic laws ; self-similar fragmentation

Date: 2004-02-13

Prépublication numéro: PMA-882