WIAS Preprint No. 979, (2004)

Alpha-stable branching and beta-coalescents



Authors

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

2010 Mathematics Subject Classification

  • 60J80 60J70 60J25 60G09 60G52 92D25

Keywords

  • Alpha-stable branching, coalescent, genealogy, lookdown construction

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)

Appeared in

  • Electron. J. Probab., 10 (9), 303-325 (electronic)

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