Fragmentation Processes with an Initial Mass Converging to Infinity

Bénédicte Haas ()
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Bénédicte Haas: CEREMADE, Université Paris-Dauphine

Journal of Theoretical Probability, 2007, vol. 20, issue 4, 721-758

Abstract: Abstract We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let F 1 (m) (t),F 2 (m) (t),… denote the decreasing rearrangement of the masses present at time t in a such process, starting from an initial mass m. Let then m→∞. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence (F 2 (m) ,F 3 (m) ,…) converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of m−F 1 (m) to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to ∞ will converge to a tree with a spine coding a fragmentation with immigration.

Keywords: Fragmentation; Immigration; Weak convergence; Regular variation; Continuum random tree; 60J25; 60F05 (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-007-0120-z

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