Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**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