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March 2016 Compensated fragmentation processes and limits of dilated fragmentations
Jean Bertoin
Ann. Probab. 44(2): 1254-1284 (March 2016). DOI: 10.1214/14-AOP1000

Abstract

A new class of fragmentation-type random processes is introduced, in which, roughly speaking, the accumulation of small dislocations which would instantaneously shatter the mass into dust, is compensated by an adequate dilation of the components. An important feature of these compensated fragmentations is that the dislocation measure $\nu$ which governs their evolutions has only to fulfill the integral condition $\int_{\mathcal{P}}(1-p_{1})^{2}\nu({\mathrm{d}}{\mathbf{p}})<\infty$, where ${\mathbf{p}}=(p_{1},\ldots)$ denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $\int_{\mathcal{P}}(1-p_{1})\nu({\mathrm{d}}{\mathbf{p}})<\infty$ for $\nu$ to be the dislocation measure of a homogeneous fragmentation. Our main results show that such compensated fragmentations naturally arise as limits of homogeneous dilated fragmentations, and bear close connections to spectrally negative Lévy processes.

Citation

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Jean Bertoin. "Compensated fragmentation processes and limits of dilated fragmentations." Ann. Probab. 44 (2) 1254 - 1284, March 2016. https://doi.org/10.1214/14-AOP1000

Information

Received: 1 April 2014; Revised: 1 December 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1344.60033
MathSciNet: MR3474471
Digital Object Identifier: 10.1214/14-AOP1000

Subjects:
Primary: 60F17, 60G51, 60G80

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 2 • March 2016
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