JournalsjemsVol. 5, No. 4pp. 395–416

The asymptotic behavior of fragmentation processes

  • Jean Bertoin

    Universität Zürich, Switzerland
The asymptotic behavior of fragmentation processes cover
Download PDF

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 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.

Cite this article

Jean Bertoin, The asymptotic behavior of fragmentation processes. J. Eur. Math. Soc. 5 (2003), no. 4, pp. 395–416

DOI 10.1007/S10097-003-0055-3