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Asymptotics of the Allele Frequency Spectrum Associated with the Bolthausen-Sznitman Coalescent

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1.	Title	Title of document	Asymptotics of the Allele Frequency Spectrum Associated with the Bolthausen-Sznitman Coalescent
2.	Creator	Author's name, affiliation, country	Anne-Laure Basdevant; Université Paul Sabatier (Toulouse III)
2.	Creator	Author's name, affiliation, country	Christina Goldschmidt; Department of Statistics, University of Oxford
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	Abstract	We consider a coalescent process as a model for the genealogy of a sample from a population. The population is subject to neutral mutation at constant rate $\rho$ per individual and every mutation gives rise to a completely new type. The allelic partition is obtained by tracing back to the most recent mutation for each individual and grouping together individuals whose most recent mutations are the same. The allele frequency spectrum is the sequence $(N_1(n), N_2(n), \ldots, N_n(n))$, where $N_k(n)$ is number of blocks of size $k$ in the allelic partition with sample size $n$. In this paper, we prove law of large numbers-type results for the allele frequency spectrum when the coalescent process is taken to be the Bolthausen-Sznitman coalescent. In particular, we show that $n^{-1}(\log n) N_1(n) {\stackrel{p}{\rightarrow}} \rho$ and, for $k \geq 2$, $n^{-1}(\log n)^2 N_k(n) {\stackrel{p}{\rightarrow}} \rho/(k(k-1))$ as $n \to \infty$. Our method of proof involves tracking the formation of the allelic partition using a certain Markov process, for which we prove a fluid limit.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2008-03-31
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/494
10.	Identifier	Digital Object Identifier	10.1214/EJP.v13-494
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 13
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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