The Annals of Probability

Beta-coalescents and continuous stable random trees

Julien Berestycki, Nathanaël Berestycki, and Jason Schweinsberg

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Abstract

Coalescents with multiple collisions, also known as Λ-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Λ is the Beta (2−α, α) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to the recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly–Kurtz lookdown process using continuous random trees, which is of independent interest. This produces a number of results concerning the small-time behavior of Beta-coalescents. Most notably, we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics.

Article information

Source
Ann. Probab. Volume 35, Number 5 (2007), 1835-1887.

Dates
First available in Project Euclid: 5 September 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1189000930

Digital Object Identifier
doi:10.1214/009117906000001114

Mathematical Reviews number (MathSciNet)
MR2349577

Zentralblatt MATH identifier
1129.60067

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60K99: None of the above, but in this section 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Coalescent with multiple collisions stable continuous random trees Galton–Watson processes multifractal spectrum frequency spectrum lookdown construction Lévy processes

Citation

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Beta-coalescents and continuous stable random trees. Ann. Probab. 35 (2007), no. 5, 1835--1887. doi:10.1214/009117906000001114. http://projecteuclid.org/euclid.aop/1189000930.


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