Two Coalescents Derived from the Ranges of Stable Subordinators

Jean Bertoin (Université Paris VI)
Jim Pitman (University of California, Berkeley)


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 Bolthausen and Sznitman.

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Pages: 1-17

Publication Date: November 10, 1999

DOI: 10.1214/EJP.v5-63


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