Alpha-Stable Branching and Beta-Coalescents

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 α-stable branching mechanisms. The random ancestral partition is then a time-changed Λcoalescent, where Λ is the Beta-distribution with parameters 2 − α and α, and the time change is given by Z 1−α , where Z is the total population size. For α = 2 (Feller's branching diffusion) and Λ = δ 0 (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem. For α = 1 and Λ 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.

Coalescent Processes When the Distribution of Offspring Number Among Individuals Is Highly Skewed

“…The probability H i is similar to mG i,1 , but requires that the single line is one of the u ÿ 1 offspring and not the parent itself. The usual way to measure time in these models is to scale it by the inverse of the coalescence probability, so that one unit of time in the limit process is equal to 1/ G 2,2 steps in the discrete-time model (Pitman 1999;Sagitov 1999;Mö hle and Sagitov 2001;Birkner et al 2005). However, as is illustrated below, we find that this choice of timescale makes it difficult to interpret the relative sizes of gene genealogies.…”

“…The limit process follows from the existence of limits (Pitman 1999;Sagitov 1999). Note that P U * (u) corresponds to the measure L invoked in other works (Pitman 1999;Sagitov 1999;Mö hle and Sagitov 2001;Birkner et al 2005).…”

“…It is the scaled family size, or the scaled number of offspring, of a large reproduction event, measured as a fraction of the total population. In comparison, work on general multiplemergers coalescent processes occurs in a more abstract mathematical setting (Pitman 1999;Sagitov 1999;Birkner et al 2005). More easily interpreted models include the power-law distribution function for family sizes that yields the beta-coalescent (Schweinsberg 2003;Birkner et al 2005) and the models of recurrent selective sweeps (Gillespie 2000) that are best approximated by a coalescent with simultaneous multiple mergers (Durrett and Schweinsberg 2005).…”

“…In comparison, work on general multiplemergers coalescent processes occurs in a more abstract mathematical setting (Pitman 1999;Sagitov 1999;Birkner et al 2005). More easily interpreted models include the power-law distribution function for family sizes that yields the beta-coalescent (Schweinsberg 2003;Birkner et al 2005) and the models of recurrent selective sweeps (Gillespie 2000) that are best approximated by a coalescent with simultaneous multiple mergers (Durrett and Schweinsberg 2005). Note that our assumption in Equation 7, that the family size of large families is on the order of the population size, is required to produce an ancestral process that is different from Kingman's coalescent given our modified Moran model; see Mö hle and Sagitov (2001Sagitov ( , p. 1552.…”

Probabilités et biologie

“…The Beta n-coalescents can be divided into three classes with 0 < α < 1, α = 1, 1 < α < 2, depending on their different types of asymptotic tree shapes as n goes to infinity. These coalescents appear in context of the super-critical Galton-Watson process [54], of continuous state branching process [10] and of continuous random trees [7]. In particular, if α = 1, we call Π (n) the Bolthausen-Sznitman n-coalescent; it appears in the field of spin glasses [12,13] and is connected to random recursive trees [29].…”