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Asymptotic laws for nonconservative self-similar fragmentations

Jean Bertoin·Alexander Gnedin
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Abstract

We consider a self-similar fragmentation process in which the generic particle of size $x$ is replaced at probability rate $x^α$, by its offspring made of smaller particles, where $α$ 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 $t^{-1/α}$ 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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Jean Bertoin, Alexander Gnedin. 2004-06-21. Asymptotic laws for nonconservative self-similar fragmentations. https://arxiv.org/abs/math/0402227

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