Two Coalescents Derived from the Ranges of Stable Subordinators
Jean Bertoin, Université Paris VI
Jim Pitman, University of California, Berkeley
Abstract
Let $M_alpha$ be the closure of the range of a stable subordinator of
index $alphain ]0,1[$.
There are two natural constructions of the $M_{alpha}$'s simultaneously for
all $alphain ]0,1[$, 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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