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Splitting trees with neutral Poissonian mutations I: Small families

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  • Champagnat, Nicolas
  • Lambert, Amaury

Abstract

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree [9], and the population counting process (Nt;t≥0) is a homogeneous, binary Crump–Mode–Jagers process.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:3:p:1003-1033
    DOI: 10.1016/j.spa.2011.11.002
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    References listed on IDEAS

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    1. 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.
    2. Geiger, Jochen, 1996. "Size-biased and conditioned random splitting trees," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 187-207, December.
    3. Jagers, Peter & Nerman, Olle, 1984. "Limit theorems for sums determined by branching and other exponentially growing processes," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 47-71, May.
    4. Durrett, Richard & Moseley, Stephen, 2010. "Evolution of resistance and progression to disease during clonal expansion of cancer," Theoretical Population Biology, Elsevier, vol. 77(1), pages 42-48.
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    Cited by:

    1. Henry, Benoit, 2021. "Approximation of the allelic frequency spectrum in general supercritical branching populations," Stochastic Processes and their Applications, Elsevier, vol. 132(C), pages 192-225.
    2. Champagnat, Nicolas & Lambert, Amaury, 2013. "Splitting trees with neutral Poissonian mutations II: Largest and oldest families," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1368-1414.
    3. Wiuf, Carsten, 2018. "Some properties of the conditioned reconstructed process with Bernoulli sampling," Theoretical Population Biology, Elsevier, vol. 122(C), pages 36-45.

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