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Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent

Author

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  • Drmota, Michael
  • Iksanov, Alex
  • Moehle, Martin
  • Roesler, Uwe

Abstract

We study the total branch length Ln of the Bolthausen-Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are applied to derive a corresponding limiting law for the total number of mutations for the Bolthausen-Sznitman coalescent with mutation rate r>0. Moreover, the results show that, for the Bolthausen-Sznitman coalescent, the total branch length Ln is closely related to Xn, the number of collision events that take place until there is just a single block. The proofs are mainly based on an analysis of random recursive equations using associated generating functions.

Suggested Citation

  • Drmota, Michael & Iksanov, Alex & Moehle, Martin & Roesler, Uwe, 2007. "Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1404-1421, October.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:10:p:1404-1421
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    Cited by:

    1. Arbisser, Ilana M. & Jewett, Ethan M. & Rosenberg, Noah A., 2018. "On the joint distribution of tree height and tree length under the coalescent," Theoretical Population Biology, Elsevier, vol. 122(C), pages 46-56.
    2. Möhle, M., 2010. "Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent," Stochastic Processes and their Applications, Elsevier, vol. 120(11), pages 2159-2173, November.
    3. Eldon, Bjarki, 2011. "Estimation of parameters in large offspring number models and ratios of coalescence times," Theoretical Population Biology, Elsevier, vol. 80(1), pages 16-28.

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