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:**

- 60G18 Self-similar processes
- 60J25 Markov processes with continuous parameter

**Résumé:** Let $(Y(t), t \geq 0)$ be the fragmentation process introduced by
Aldous and Pitman that can be obtained by time-reversing the
standard additive coalescent.
Let $(\sigma_{1/2}(t), t \geq 0)$ be the stable subordinator of
index $1/2$. Aldous and Pitman showed that the distribution of the
sizes of the fragments of $Y(t)$ is the same as the conditional
distribution of the jump sizes of $\sigma_{1/2}$ up to time $t$,
given $\sigma_{1/2}(t) = 1$. We show that this is a special property
of the stable subordinator of index $1/2$, in the sense that if
$\alpha \neq 1/2$ and $\sigma_{\alpha}$ is the stable subordinator
of index $\alpha$, then there exists no self-similar fragmentation
for which the distribution of the sizes of the fragments at time $t$
equals the conditional distribution of the jump sizes of
$\sigma_{\alpha}$ up to time $t$, given $\sigma_{\alpha}(t) = 1$.
We also show that a property relating the distribution of a size-biased
pick from $Y(t)$ to the distribution of $\sigma_{1/2}(t)$
is similarly particular to the $\alpha = 1/2$ case. However, we show
that for each $\alpha \in (0,1)$, there is a family of
self-similar fragmentations whose behavior as $t \downarrow 0$ is
related to the stable subordinator of index $\alpha$ in the same way
that the behavior of $Y(t)$ as $t \downarrow 0$ is related to the
stable subordinator of index $1/2$.

**Mots Clés:** *Self-similar fragmentation ; stable subordinator ; Poisson-Kingman distribution*

**Date:** 2002-05-16

**Prépublication numéro:** *PMA-726*

**Pdf file : **PMA-726.pdf