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Splitting trees with neutral Poissonian mutations II: Largest and oldest families

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  • Champagnat, Nicolas
  • Lambert, Amaury

Abstract

We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:4:p:1368-1414
    DOI: 10.1016/j.spa.2012.11.013
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    References listed on IDEAS

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    1. 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.
    2. Pardis C. Sabeti & David E. Reich & John M. Higgins & Haninah Z. P. Levine & Daniel J. Richter & Stephen F. Schaffner & Stacey B. Gabriel & Jill V. Platko & Nick J. Patterson & Gavin J. McDonald & Han, 2002. "Detecting recent positive selection in the human genome from haplotype structure," Nature, Nature, vol. 419(6909), pages 832-837, October.
    3. Geiger, Jochen, 1996. "Size-biased and conditioned random splitting trees," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 187-207, December.
    4. 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.
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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.

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