The total external length of the evolving Kingman coalescent

Abstract

The evolving Kingman coalescent is the tree-valued process which records the time evolution undergone by the genealogies of Moran populations. We consider the associated process of total external tree length of the evolving Kingman coalescent and its asymptotic behaviour when the number of leaves of the tree tends to infinity. We show that on the time-scale of the Moran model slowed down by a factor equal to the population size, the (centred and rescaled) external length process converges to a stationary Gaussian process with almost surely continuous paths and covariance function \(c(s,t)=\Big ( \frac{2}{2+|s-t|} \Big )^2\). A key role in the evolution of the external length is played by the internal lengths of finite orders in the coalescent at a fixed time which behave asymptotically in a multivariate Gaussian manner [see Dahmer and Kersting (Ann Appl Probab 25(3):1325–1348, 2015)]. A coupling of the Moran model with a critical branching process is used. We also derive a central limit result for normally distributed sums endowed with independent random coefficients.

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Acknowledgments

We thank the referees for very careful reading and for their hints which led to an improvement of the paper.

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Correspondence to Iulia Dahmer.

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Work partially supported by the Deutsche Forschungsgemeinschaft (DFG) Priority Programme SPP 1590 “Probabilistic Structures in Evolution”. ID was partially supported by the German Academic Exchange Service (DAAD).

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Dahmer, I., Kersting, G. The total external length of the evolving Kingman coalescent. Probab. Theory Relat. Fields 167, 1165–1214 (2017). https://doi.org/10.1007/s00440-016-0703-7

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Keywords

  • Evolving Kingman coalescent
  • External length process
  • Gaussian process
  • Coupling
  • Critical branching process

Mathematics Subject Classification

  • 60K35
  • 60F05
  • 60J10