Two Coalescents Derived from the Ranges of Stable Subordinators

Let M� be the closure of the range of a stable subordinator of index ��]0,1[. There are two natural constructions of the M�’s simultaneously for all ��]0,1[, so that M��M� for 0<�<�<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−� for 0<�<1. In particular, we identify the coalescent based on the subordination scheme with the coalescent recently introduced by Bolthausen and Sznitman. DOI: 10.1214/EJP.v5-63 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-79476 Published Version Originally published at: Bertoin, Jean; Pitman, Jim (2000). Two coalescents derived from the ranges of stable subordinators. Electronic Journal of Probability, 5(7):1-17. DOI: 10.1214/EJP.v5-63 E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol. 5 (2000) Paper no. 7, pages 1–17. Journal URL http://www.math.washington.edu/~ejpecp/ Paper URL http://www.math.washington.edu/~ejpecp/EjpVol5/paper7.abs.html TWO COALESCENTS DERIVED FROM THE RANGES OF STABLE SUBORDINATORS Jean Bertoin Laboratoire de Probabilités Tour 56, 4 Place Jussieu, 75252 Paris Cedex 05, France [email protected] Jim Pitman Department of Statistics, University of California, Berkeley 367 Evans Hall # 3860 Berkeley, CA 94720-3860, USA. [email protected] 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 [6].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 [6].