TY - JOUR
A1 - Birkner, Matthias
A1 - Blath, Jochen
A1 - Capaldo, Marcella
A1 - Etheridge, Alison
A1 - MÃ¶hle, Martin
A1 - Schweinsberg, Jason
A1 - Wakolbinger, Anton
T1 - Alpha-stable branching and beta-coalescents
T2 - Electronic journal of probability
N2 - 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.
KW - alpha-stable branching
KW - coalescent
KW - genealogy
KW - lookdown construction
Y1 - 2005
UR - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/32891
UR - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-328911
SN - 1083-6489
VL - 10
SP - 303
EP - 325
PB - EMIS ELibEMS
CY - [Madralin]
ER -