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 (m)1 (t),F (m)2 (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 (m)2 ,F (m)3 ,…) converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of m−F (m)1 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.
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Research supported in part by EPSRC GR/T26368.
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Haas, B. Fragmentation Processes with an Initial Mass Converging to Infinity. J Theor Probab 20, 721–758 (2007). https://doi.org/10.1007/s10959-007-0120-z
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DOI: https://doi.org/10.1007/s10959-007-0120-z