# The asymptotic behavior of fragmentation processes

Journal of the European Mathematical Society (2003)

- Volume: 005, Issue: 4, page 395-416
- ISSN: 1435-9855

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topBertoin, Jean. "The asymptotic behavior of fragmentation processes." Journal of the European Mathematical Society 005.4 (2003): 395-416. <http://eudml.org/doc/277459>.

@article{Bertoin2003,

abstract = {The fragmentation processes considered in this work are self-similar Markov
processes which are meant to describe the evolution of a mass that falls apart randomly as time passes. We investigate their pathwise asymptotic behavior as $t\rightarrow \infty $. In the so-called homogeneous case, we first point at a law of large numbers and a central limit theorem for (a modified version of) the empirical distribution of the fragments at time $t$. These results are reminiscent of those of Asmussen and Kaplan [3] and Biggins [12] for branching random walks. Next, in the same vein as Biggins [10], we also investigate some natural martingales, which open the way to an almost sure large deviation principle by an application of the
Gärtner-Ellis theorem. Finally, some asymptotic results in the general self-similar case are derived by time-change from the previous ones. Properties of size-biased picked fragments
provide key tools for the study.},

author = {Bertoin, Jean},

journal = {Journal of the European Mathematical Society},

keywords = {fragmentation; self-similar; central limit theorem; large deviations; fragmentation; self-similar; central limit theorem; large deviations},

language = {eng},

number = {4},

pages = {395-416},

publisher = {European Mathematical Society Publishing House},

title = {The asymptotic behavior of fragmentation processes},

url = {http://eudml.org/doc/277459},

volume = {005},

year = {2003},

}

TY - JOUR

AU - Bertoin, Jean

TI - The asymptotic behavior of fragmentation processes

JO - Journal of the European Mathematical Society

PY - 2003

PB - European Mathematical Society Publishing House

VL - 005

IS - 4

SP - 395

EP - 416

AB - The fragmentation processes considered in this work are self-similar Markov
processes which are meant to describe the evolution of a mass that falls apart randomly as time passes. We investigate their pathwise asymptotic behavior as $t\rightarrow \infty $. In the so-called homogeneous case, we first point at a law of large numbers and a central limit theorem for (a modified version of) the empirical distribution of the fragments at time $t$. These results are reminiscent of those of Asmussen and Kaplan [3] and Biggins [12] for branching random walks. Next, in the same vein as Biggins [10], we also investigate some natural martingales, which open the way to an almost sure large deviation principle by an application of the
Gärtner-Ellis theorem. Finally, some asymptotic results in the general self-similar case are derived by time-change from the previous ones. Properties of size-biased picked fragments
provide key tools for the study.

LA - eng

KW - fragmentation; self-similar; central limit theorem; large deviations; fragmentation; self-similar; central limit theorem; large deviations

UR - http://eudml.org/doc/277459

ER -

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- Bénédicte Haas, Regularity of formation of dust in self-similar fragmentations
- M. Escobedo, S. Mischler, M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models
- Robert Knobloch, Andreas E. Kyprianou, Survival of homogeneous fragmentation processes with killing
- Christina Goldschmidt, Bénédicte Haas, Behavior near the extinction time in self-similar fragmentations I : the stable case
- S. C. Harris, R. Knobloch, A. E. Kyprianou, Strong law of large numbers for fragmentation processes

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