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)


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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Pages: 303-325

Publication Date: March 4, 2005

DOI: 10.1214/EJP.v10-241


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