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

### On small masses in self-similar fragmentations

Auteur(s):

Code(s) de Classification MSC:

• 60J25 Markov processes with continuous parameter
• 60F15 Strong theorems

Résumé: We consider a self-similar fragmentation process which preserves the total mass. We are interested in the asymptotic behavior as $\varepsilon\to0+$ of $N(\varepsilon,t)={\tt Card}\left\{i: X_i(t)>\varepsilon\right\}$, the number of the fragments with size greater than $\varepsilon$ at some fixed time $t>0$. Under a certain condition of regular variation type on the so-called dislocation measure, we exhibit a deterministic function $\varphi:]0,1[\to]0,\infty[$ such that the limit of $N(\varepsilon,t)/\varphi(\varepsilon)$ exists and is non-degenerate. In general the limit is random, but may be deterministic when a certain relation between the index of self-similarity and the dislocation measure holds. We also present a similar result for the total mass of fragments less than $\varepsilon$.

Mots Clés: fragmentation ; self-similar ; strong limit theorems

Date: 2002-07-03

Prépublication numéro: PMA-745

Pdf file : PMA-745.pdf