# Asymptotic laws for nonconservative self-similar fragmentations

Bertoin, J; Gnedin, A (2004). Asymptotic laws for nonconservative self-similar fragmentations. Electronic Journal of Probability, 9(19):575-593.

## Abstract

We consider a self-similar fragmentation process in which the generic particle of size $x$ is replaced at probability rate $x^\alpha$, by its offspring made of smaller particles, where $\alpha$ is some positive parameter. The total of offspring sizes may be both larger or smaller than $x$ with positive probability. We show that under certain conditions the typical size in the ensemble is of the order $t^{-1/\alpha}$ and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law.

## Abstract

We consider a self-similar fragmentation process in which the generic particle of size $x$ is replaced at probability rate $x^\alpha$, by its offspring made of smaller particles, where $\alpha$ is some positive parameter. The total of offspring sizes may be both larger or smaller than $x$ with positive probability. We show that under certain conditions the typical size in the ensemble is of the order $t^{-1/\alpha}$ and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2004 26 Jun 2013 16:06 19 Feb 2018 20:42 Institute of Mathematical Statistics 1083-6489 Closed Publisher DOI. An embargo period may apply. https://doi.org/10.1214/EJP.v9-215 http://www.ams.org/mathscinet-getitem?mr=2080610http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1064.60075