Geodesic
The asymptotic behavior of fragmentation processes
Journal of the European Mathematical Society, Tome 5 (2003) no. 4, pp. 395-416 Cet article a éte moissonné depuis la source EMS Press

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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 foes to infinity. 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.
DOI : 10.1007/s10097-003-0055-3
Classification : 60-XX, 00-XX
Keywords: fragmentation, self-similar, central limit theorem, large deviations 
@article{JEMS_2003_5_4_a2,
     author = {Jean Bertoin},
     title = {The asymptotic behavior of fragmentation processes},
     journal = {Journal of the European Mathematical Society},
     pages = {395--416},
     year = {2003},
     volume = {5},
     number = {4},
     doi = {10.1007/s10097-003-0055-3},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10097-003-0055-3/}
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Jean Bertoin. The asymptotic behavior of fragmentation processes. Journal of the European Mathematical Society, Tome 5 (2003) no. 4, pp. 395-416. doi: 10.1007/s10097-003-0055-3

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