%0 Journal Article %1 stable-branching-coalescing %A Birkner, Matthias %A Blath, Jochen %A Capaldo, Marcella %A Etheridge, Alison M. %A Möhle, Martin %A Schweinsberg, Jason %A Wakolbinger, Anton %B Alpha-Stable Branching and Beta-Coalescents %D 2005 %J Electronic Journal of Probability %K coalescent_theory continuous-state_branching_processes lambda_coalescent stable_processes %N 9 %T Alpha-Stable Branching and Beta-Coalescents %U http://ejp.ejpecp.org/article/view/241 %V 10 %X 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.

@article{stable-branching-coalescing, 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); the other is based on direct calculations with generators. }, added-at = {2009-01-22T17:17:46.000+0100}, author = {Birkner, Matthias and Blath, Jochen and Capaldo, Marcella and Etheridge, Alison M. and Möhle, Martin and Schweinsberg, Jason and Wakolbinger, Anton}, biburl = {https://www.bibsonomy.org/bibtex/273710426ce7d7c120df9afd6f4c04ee2/peter.ralph}, booktitle = {Alpha-Stable Branching and Beta-Coalescents}, description = {Electronic Journal of Probability - Vol. 10 (2005)}, interhash = {fc7c34b68b5a4d711b3b0b3883f22acd}, intrahash = {73710426ce7d7c120df9afd6f4c04ee2}, journal = {Electronic Journal of Probability}, keywords = {coalescent_theory continuous-state_branching_processes lambda_coalescent stable_processes}, number = 9, timestamp = {2013-12-14T06:26:09.000+0100}, title = {Alpha-Stable Branching and Beta-Coalescents}, url = {http://ejp.ejpecp.org/article/view/241}, volume = 10, year = 2005 }