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On small masses in self-similar fragmentations


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.

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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11 citations in Web of Science®
11 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
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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