Asymptotic Laws for Nonconservative Self-similar Fragmentations
Jean Bertoin, Université Paris VI
Alexander V. Gnedin, Rijksuniversiteit Utrecht, The Netherlands
Abstract
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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