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A limit theorem for trees of alleles in branching processes with rare neutral mutations

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  • Bertoin, Jean

Abstract

We are interested in the genealogical structure of alleles for a Bienaymé-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Jirina process) in discrete time. Itô's excursion theory and the Lévy-Itô decomposition of subordinators provide fundamental insights for the results.

Suggested Citation

  • Bertoin, Jean, 2010. "A limit theorem for trees of alleles in branching processes with rare neutral mutations," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 678-697, May.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:5:p:678-697
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    References listed on IDEAS

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    1. Le Gall, Jean-François, 2010. "Itô's excursion theory and random trees," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 721-749, May.
    2. Watanabe, Shinzo, 2010. "Itô's theory of excursion point processes and its developments," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 653-677, May.
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    Cited by:

    1. Chen, Xinxin, 2013. "Convergence rate of the limit theorem of a Galton–Watson tree with neutral mutations," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 588-595.
    2. Normand, Raoul, 2014. "Two population models with constrained migrations," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1773-1812.
    3. Champagnat, Nicolas & Lambert, Amaury, 2012. "Splitting trees with neutral Poissonian mutations I: Small families," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1003-1033.

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