The asymptotic behavior of fragmentation processes |
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Authors: | Email author" target="_blank">Jean?BertoinEmail author |
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Institution: | (1) Laboratoire de Probabilités et Modèles Aléatoires et Institut universitaire de France, Université Pierre et Marie Curie, et C.N.R.S. UMR 7599, 175, rue du Chevaleret, F-75013 Paris, France |
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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![rarr](/content/x8m19r4g1xrdm6m5/xxlarge8594.gif) . 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. Mathematics Subject Classification (2000) 60J25, 60G09 |
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Keywords: | fragmentation self-similar central limit theorem large deviations |
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