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The asymptotic behavior of fragmentation processes

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Journal of the European Mathematical Society

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→∞. 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.

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Authors and Affiliations

  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

    Jean Bertoin

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  1. Jean Bertoin
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Correspondence to Jean Bertoin.

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Mathematics Subject Classification (2000)

60J25, 60G09

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Bertoin, J. The asymptotic behavior of fragmentation processes. J. Eur. Math. Soc. 5, 395–416 (2003). https://doi.org/10.1007/s10097-003-0055-3

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  • DOI: https://doi.org/10.1007/s10097-003-0055-3

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