## Two coalescents derived from the ranges of stable subordinators.(English)Zbl 0949.60034

Summary: Let $$M_\alpha$$ be the closure of the range of a stable subordinator of index $$\alpha\in ]0,1[$$. There are two natural constructions of the $$M_{\alpha}$$’s simultaneously for all $$\alpha\in ]0,1[$$, so that $$M_{\alpha}\subseteq M_{\beta}$$ for $$0< \alpha < \beta < 1$$: one based on the intersection of independent regenerative sets and one based on Bochner’s subordination. We compare the corresponding two coalescent processes defined by the lengths of complementary intervals of $$[0,1]\backslash M_{1-\rho}$$ for $$0 < \rho < 1$$. In particular, we identify the coalescent based on the subordination scheme with the coalescent recently introduced by E. Bolthausen and A.-S. Sznitman [Commun. Math. Phys. 197, No. 2, 247-276 (1998; Zbl 0927.60071)].

### MSC:

 60E07 Infinitely divisible distributions; stable distributions 60G57 Random measures

Zbl 0927.60071
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