Alpha-Stable Branching and Beta-Coalescents
Matthias Birkner, Weierstrass Institute for Applied Analysis and Stochastics, Germany
Jochen Blath, University of Oxford, UK
Marcella Capaldo, University of Oxford, UK
Alison M. Etheridge, University of Oxford, UK
Martin Möhle, University of Tübingen, Germany
Jason Schweinsberg, University of California at San Diego, USA
Anton Wakolbinger, J. W. Goethe Universität
Abstract
We determine that the continuous-state branching
processes for which the genealogy, suitably time-changed, can be
described by an autonomous Markov process are precisely those arising
from $alpha$-stable branching mechanisms. The random ancestral
partition is then a time-changed $Lambda$-coalescent, where $Lambda$
is the Beta-distribution with parameters $2-alpha$ and $alpha$, and
the time change is given by $Z^{1-alpha}$, where $Z$ is the total
population size. For $alpha = 2$ (Feller's branching diffusion) and
$Lambda = delta_0$ (Kingman's coalescent), this is in the spirit of
(a non-spatial version of) Perkins' Disintegration Theorem. For
$alpha =1$ and $Lambda$ the uniform distribution on $[0,1]$, this is
the duality discovered by Bertoin & Le Gall (2000) between the
norming of Neveu's continuous state branching process and the
Bolthausen-Sznitman coalescent.
We present two approaches: one, exploiting the `modified lookdown
construction', draws heavily on Donnelly & Kurtz (1999); the other
is based on direct calculations with generators.
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