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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60J25 Markov processes with continuous parameter
- 60G09 Exchangeability

**Résumé:** The fragmentation processes considered in this work are self-similar
Markov processes which are meant to describe the ranked sequence
of the masses of the pieces of an object that falls apart randomly as
time passes.
We investigate their behavior as $t\to\infty$. Roughly, we show that
the rate of decay of the $\ell^p$-norm (where $p>1$) is exponential
when the index of self-similarity $\alpha$ is $0$, polynomial when
$\alpha>0$,
whereas the entire mass disappears in a finite time when $\alpha<0$.
Moreover, we establish a strong limit theorem for the empirical measure
of the fragments in the case when $\alpha>0$. Properties of size-biased
picked fragments provide key tools for the study.

**Mots Clés:** *Fragmentation ; self-similar ; scattering rate ; empirical measure*

**Date:** 2001-04-25

**Prépublication numéro:** *PMA-651*