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

Bertoin, Jean; Pitman, Jim

Electron. J. Probab. 5, Paper No. 7, 17 p. (2000).

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

Keywords:

coalescent; stable subordinator; Poisson-Dirichlet distribution

Citations:

Zbl 0927.60071