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

### Self-similar fragmentations and stable subordinators

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