Electronic Journal of Probability

Alpha-Stable Branching and Beta-Coalescents

Matthias Birkner, Jochen Blath, Marcella Capaldo, Alison Etheridge, Martin Möhle, Jason Schweinsberg, and Anton Wakolbinger

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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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Electron. J. Probab., Volume 10 (2005), paper no. 9, 303-325.

Accepted: 4 March 2005
First available in Project Euclid: 1 June 2016

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Birkner, Matthias; Blath, Jochen; Capaldo, Marcella; Etheridge, Alison; Möhle, Martin; Schweinsberg, Jason; Wakolbinger, Anton. Alpha-Stable Branching and Beta-Coalescents. Electron. J. Probab. 10 (2005), paper no. 9, 303--325. doi:10.1214/EJP.v10-241. https://projecteuclid.org/euclid.ejp/1464816810

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