Asymptotic Laws for Nonconservative Self-similar Fragmentations

Jean Bertoin (Université Paris VI)
Alexander V. Gnedin (Rijksuniversiteit Utrecht, The Netherlands)


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

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Pages: 575-593

Publication Date: July 30, 2004

DOI: 10.1214/EJP.v9-215


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