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The tree length of an evolving coalescent
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  • Published: 25 June 2010

The tree length of an evolving coalescent

  • P. Pfaffelhuber1,
  • A. Wakolbinger2 &
  • H. Weisshaupt3 

Probability Theory and Related Fields volume 151, pages 529–557 (2011)Cite this article

  • 239 Accesses

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Abstract

A well-established model for the genealogy of a large population in equilibrium is Kingman’s coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the population size tends to infinity, to a limit process with càdlàg paths, infinite infinitesimal variance, and a Gumbel distribution as its equilibrium.

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Author information

Authors and Affiliations

  1. Abteilung Mathematische Stochastik, Albert-Ludwigs Universität, Eckerstr. 1, 79104, Freiburg, Germany

    P. Pfaffelhuber

  2. Institut für Mathematik, Goethe-Universität, Robert-Mayer-Str. 10, 60325, Frankfurt, Germany

    A. Wakolbinger

  3. Center for Biosystems Analysis, Albert-Ludwigs Universität, Habsburger Str. 49, Freiburg, Germany

    H. Weisshaupt

Authors
  1. P. Pfaffelhuber
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  2. A. Wakolbinger
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  3. H. Weisshaupt
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Corresponding author

Correspondence to P. Pfaffelhuber.

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Pfaffelhuber, P., Wakolbinger, A. & Weisshaupt, H. The tree length of an evolving coalescent. Probab. Theory Relat. Fields 151, 529–557 (2011). https://doi.org/10.1007/s00440-010-0307-6

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  • Received: 17 August 2009

  • Revised: 16 May 2010

  • Published: 25 June 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0307-6

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Keywords

  • Kingman’s coalescent
  • Genealogical trees
  • Moran model
  • Evolution of tree length
  • Large population limit
  • Gumbel distribution

Mathematics Subject Classification (2000)

  • Primary 60K35
  • Secondary 92D25
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