The Annals of Probability

Compensated fragmentation processes and limits of dilated fragmentations

Jean Bertoin

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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.

Article information

Ann. Probab. Volume 44, Number 2 (2016), 1254-1284.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G51: Processes with independent increments; Lévy processes 60G80

Homogeneous fragmentation dilation compensation dislocation measure


Bertoin, Jean. Compensated fragmentation processes and limits of dilated fragmentations. Ann. Probab. 44 (2016), no. 2, 1254--1284. doi:10.1214/14-AOP1000.

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