Two coalescents derived from the ranges of stable subordinators

Abstract

Let Mα be the closure of the range of a stable subordinator of index α∈]0,1[. There are two natural constructions of the Mα's simultaneously for all α∈]0,1[, so that Mα⊆Mβ for 0<α<β<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]∖M1−ρ for 0<ρ<1. In particular, we identify the coalescent based on the subordination scheme with the coalescent recently introduced by Bolthausen and Sznitman.