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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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2004
Deposited On:26 Jun 2013 16:06
Last Modified:06 Nov 2017 10:19
Publisher:Institute of Mathematical Statistics
ISSN:1083-6489
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/EJP.v9-215
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2080610
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1064.60075

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