Abstract
Let (Y(t), t ≥ 0) be the fragmentation process introduced by Aldous and Pitman that can be obtained by time-reversing the standard additive coalescent. Let (σ 1/2(t), t ≥ 0) be the stable subordinator of index 1/2. Aldous and Pitman showed that the distribution of the sizes of the fragments of Y(t) is the same as the conditional distribution of the jump sizes of σ 1/2 up to time t, given σ 1/2(t) = 1. We show that this is a special property of the stable subordinator of index 1/2, in the sense that if α ≠ 1/2 and σ α is the stable subordinator of index α, then there exists no self-similar fragmentation for which the distribution of the sizes of the fragments at time t equals the conditional distribution of the jump sizes of σ α up to time t, given σ α (t) = 1. We also show that a property relating the distribution of a size-biased pick from Y(t) to the distribution of σ 1/2(t) is similarly particular to the α = 1/2 case. However, we show that for each α∈(0,1), there is a family of self-similar fragmentations whose behavior as t↓0 is related to the stable subordinator of index α in the same way that the behavior of Y(t) as t↓0 is related to the stable subordinator of index 1/2.
Keywords: Self-similar fragmentation, stable subordinator, Poisson–Kingman distribution.
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© 2003 Springer-Verlag Berlin Heidelberg
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Miermont, G., Schweinsberg, J. (2003). Self-similar fragmentations and stable subordinators. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics, vol 1832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40004-2_13
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DOI: https://doi.org/10.1007/978-3-540-40004-2_13
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