# Asymptotic sampling formulae for $\mathit{\Lambda}$-coalescents

Julien Berestycki; Nathanaël Berestycki; Vlada Limic

Annales de l'I.H.P. Probabilités et statistiques (2014)

- Volume: 50, Issue: 3, page 715-731
- ISSN: 0246-0203

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topBerestycki, Julien, Berestycki, Nathanaël, and Limic, Vlada. "Asymptotic sampling formulae for $\varLambda $-coalescents." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 715-731. <http://eudml.org/doc/272053>.

@article{Berestycki2014,

abstract = {We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a $\varLambda $-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to $\infty $. Some of our results hold in the case of a general $\varLambda $-coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at $\alpha =3/2$, where $\alpha \in (1,2)$ is the exponent of regular variation.},

author = {Berestycki, Julien, Berestycki, Nathanaël, Limic, Vlada},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {$\varLambda $-coalescents; speed of coming down from infinity; exchangeable coalescents; sampling formulae; infinite allele model; genetic variation; -coalescents; exchangeable coalescents; sampling formulae; infinite allele model; population dynamics; genetic variation},

language = {eng},

number = {3},

pages = {715-731},

publisher = {Gauthier-Villars},

title = {Asymptotic sampling formulae for $\varLambda $-coalescents},

url = {http://eudml.org/doc/272053},

volume = {50},

year = {2014},

}

TY - JOUR

AU - Berestycki, Julien

AU - Berestycki, Nathanaël

AU - Limic, Vlada

TI - Asymptotic sampling formulae for $\varLambda $-coalescents

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2014

PB - Gauthier-Villars

VL - 50

IS - 3

SP - 715

EP - 731

AB - We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a $\varLambda $-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to $\infty $. Some of our results hold in the case of a general $\varLambda $-coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at $\alpha =3/2$, where $\alpha \in (1,2)$ is the exponent of regular variation.

LA - eng

KW - $\varLambda $-coalescents; speed of coming down from infinity; exchangeable coalescents; sampling formulae; infinite allele model; genetic variation; -coalescents; exchangeable coalescents; sampling formulae; infinite allele model; population dynamics; genetic variation

UR - http://eudml.org/doc/272053

ER -

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