Alpha-Stable Branching and Beta-Coalescents

By Matthias C. F. Birkner, Jochen Blath, Marcella Capaldo, A. Etheridge, Martin Möhle, Jason Schweinsberg, Anton Wakolbinger. 2005

124 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b60cf4af92320b90e8bf6f124678fe6a79a6c493

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Citations in the corpus

BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES (2006, methodology): “and let R−1(t) = inf{s: Rs > t}. It follows from [ 14 ] that R−1(t) < ∞ a.s.”
Combinatorial Stochastic Processes (2006, background): “See [396] for details, and [398, 67, 126] for futher work on how coalescents with multiple and simultaneous multiple collisions arise as limits in various discrete or continuous population models.”
Small-time behavior of beta coalescents (2006, unknown): “See also Section 2 of [11] for a review of this construction in the β = 0 case.”
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees) (2009, background): “Furthermore a -coalescent is up to time change dual to the process of relative frequencies of families of a Galton–Watson process with possibly infinite variance offspring distribution (compare [3]).”
A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks (2008, methodology): “process without mutation has the form (see [ BBC05 , Equation (1.11)])”
A New Model for Evolution in a Spatial Continuum (2009, methodology): “An explicit explanation can be found in Birkner et al. (2005).”