The asymptotic behavior of fragmentation processes

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 -