We consider a self-similar fragmentation process which preserves the total mass. We are interested in the asymptotic behavior as ε→0+ of , the number of fragments with size greater than ε 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 ϕ:]0,1[→]0,∞[ such that the limit of N(ε,t)/ϕ(ε) 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 ε.

Bertoin, Jean (2004). *On small masses in self-similar fragmentations.* Stochastic Processes and their Applications, 109(1):13-22.

## Abstract

We consider a self-similar fragmentation process which preserves the total mass. We are interested in the asymptotic behavior as ε→0+ of , the number of fragments with size greater than ε 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 ϕ:]0,1[→]0,∞[ such that the limit of N(ε,t)/ϕ(ε) 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 ε.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2004 |

Deposited On: | 26 Jun 2013 16:10 |

Last Modified: | 05 Apr 2016 16:49 |

Publisher: | Elsevier |

ISSN: | 0304-4149 |

Free access at: | Publisher DOI. An embargo period may apply. |

Publisher DOI: | https://doi.org/10.1016/j.spa.2003.08.001 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=2024841 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1075.60092 |

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