# Asymptotic sampling formulae and particle system representations for $\Lambda$-coalescents

Abstract : Consider an evolving population, with genealogy given by a Lambda-coalescent that comes down from infinity. We provide rather explicit sampling formulae under this model, for large samples. More precisely, we describe the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to infinity. A regular variation condition on the driving measure Lambda is assumed for some of the almost sure asymptotic results, but most of out results are valid for a general Lambda-coalescent that comes down from infinity. The proofs rely in part on the recent analysis of the speed of coming down from infinity for Lambda-coalescents, done by the authors in a previous work. The second goal of this paper is to investigate a remarkable connection between Lambda-coalescents and genealogies of continuous-state branching processes. Our particle representation and the resulting coupling construction offer new perspective on the speed of coming down from infinity, and its consequences, as well as several other results recently obtained in the area.
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Document type :
Preprints, Working Papers, ...
39 pages. 2011
Domain :

https://hal.archives-ouvertes.fr/hal-00554696
Contributor : Julien Berestycki <>
Submitted on : Tuesday, January 11, 2011 - 10:13:16 AM
Last modification on : Monday, May 29, 2017 - 2:24:21 PM

### Identifiers

• HAL Id : hal-00554696, version 1
• ARXIV : 1101.1875

### Citation

Julien Berestycki, Nathanaël Berestycki, Vlada Limic. Asymptotic sampling formulae and particle system representations for $\Lambda$-coalescents. 39 pages. 2011. <hal-00554696>

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