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The asymptotic behavior of fragmentation processes. (English) Zbl 1042.60042
Summary: 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 \(t\to\infty\). 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 S. Asmussen and N. Kaplan [Stochastic Processes Appl. 4, 1–13, 15–31 (1976; Zbl 0317.60034 resp. Zbl 0322.60065)] and J. D. Biggins [ibid. 34, 255–274 (1990; Zbl 0703.60083)] for branching random walks. Next in the same vein as J. D. Biggins [J. Appl. Probab. 14, 25–37 (1977; Zbl 0356.60053)], 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.

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G09 Exchangeability for stochastic processes
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