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On small masses in self-similar fragmentations

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  • Bertoin, Jean

Abstract

We consider a self-similar fragmentation process which preserves the total mass. We are interested in the asymptotic behavior as [var epsilon]-->0+ of , the number of fragments with size greater than [var epsilon] at some fixed time t>0. Under a certain condition of regular variation type on the so-called dislocation measure, we exhibit a deterministic function [phi]:]0,1[-->]0,[infinity][ such that the limit of N([var epsilon],t)/[phi] ([var epsilon]) exists and is non-degenerate. In general the limit is random, but may be deterministic when a certain relation between the index of self-similarity and the dislocation measure holds. We also present a similar result for the total mass of fragments less than [var epsilon].

Suggested Citation

  • Bertoin, Jean, 2004. "On small masses in self-similar fragmentations," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 13-22, January.
    • Handle: RePEc:eee:spapps:v:109:y:2004:i:1:p:13-22
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    References listed on IDEAS

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    1. Haas, Bénédicte, 2003. "Loss of mass in deterministic and random fragmentations," Stochastic Processes and their Applications, Elsevier, vol. 106(2), pages 245-277, August.
    2. Schweinsberg, Jason, 2001. "Applications of the continuous-time ballot theorem to Brownian motion and related processes," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 151-176, September.
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    Cited by:

    1. Delmas, Jean-François, 2007. "Fragmentation at height associated with Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 297-311, March.

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