TY  - JOUR
A1  - Birkner, Matthias
A1  - Blath, Jochen
A1  - Capaldo, Marcella
A1  - Etheridge, Alison
A1  - Möhle, 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  -