Sample genealogy and mutational patterns for critical branching populations

Abstract : We study a universal object for the genealogy of a sample in populations with mutations: the critical birth-death process with Poissonian mutations, conditioned on its population size at a fixed time horizon. We show how this process arises as the law of the genealogy of a sample in a large class of critical branching populations with mutations at birth, namely populations converging, in a large population asymptotic, towards the continuum random tree. We extend this model to populations with random foundation times, with (potentially improper) prior distributions g_i: x\mapsto x^{-i}, i\in\Z_+, including the so-called uniform (i=0) and log-uniform (i=1) priors. We first investigate the mutational patterns arising from these models, by studying the site frequency spectrum of a sample with fixed size, i.e. the number of mutations carried by k individuals in the sample. Explicit formulae for the expected frequency spectrum of a sample are provided, in the cases of a fixed foundation time, and of a uniform and log-uniform prior on the foundation time. Second, we establish the convergence in distribution, for large sample sizes, of the (suitably renormalized) tree spanned by the sample genealogy with prior g_i on the time of origin. We finally prove that the limiting genealogies with different priors can all be embedded in the same realization of a given Poisson point measure.
Type de document :
Pré-publication, Document de travail
28 pages, 6 figures. 2014
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https://hal.archives-ouvertes.fr/hal-01053343
Contributeur : Cécile Delaporte <>
Soumis le : mercredi 30 juillet 2014 - 13:29:43
Dernière modification le : mardi 11 octobre 2016 - 14:10:46

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  • HAL Id : hal-01053343, version 1
  • ARXIV : 1407.7720

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Citation

G. Achaz, C. Delaporte, A. Lambert. Sample genealogy and mutational patterns for critical branching populations. 28 pages, 6 figures. 2014. <hal-01053343>

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