The asymptotic behavior of fragmentation processes

Jean Bertoin

Journal of the European Mathematical Society (2003)

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

Abstract

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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 . 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.

How to cite

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Bertoin, 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 -

Citations in EuDML Documents

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  1. Eric Fekete, Branching random walks on binary search trees: convergence of the occupation measure
  2. M. Escobedo, S. Mischler, Dust and self-similarity for the Smoluchowski coagulation equation
  3. Bo Chen, Matthias Winkel, Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
  4. Bénédicte Haas, Regularity of formation of dust in self-similar fragmentations
  5. M. Escobedo, S. Mischler, M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models
  6. Robert Knobloch, Andreas E. Kyprianou, Survival of homogeneous fragmentation processes with killing
  7. Christina Goldschmidt, Bénédicte Haas, Behavior near the extinction time in self-similar fragmentations I : the stable case
  8. S. C. Harris, R. Knobloch, A. E. Kyprianou, Strong law of large numbers for fragmentation processes

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