Electronic Journal of Probability

The evolving beta coalescent

Götz Kersting, Jason Schweinsberg, and Anton Wakolbinger

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In mathematical population genetics, it is well known that one can represent the genealogy of a population by a tree, which indicates how the ancestral lines of individuals in the population coalesce as they are traced back in time.  As the population evolves over time, the tree that represents the genealogy of the population also changes, leading to a tree-valued stochastic process known as the evolving coalescent.  Here we will consider the evolving coalescent for populations whose genealogy can be described by a beta coalescent, which is known to give the genealogy of populations with very large family sizes.  We show that as the size of the population tends to infinity, the evolution of certain functionals of the beta coalescent, such as the total number of mergers, the total branch length, and the total length of external branches, converges to a stationary stable process.  Our methods also lead to new proofs of known asymptotic results for certain functionals of the non-evolving beta coalescent.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 64, 27 pp.

Accepted: 20 July 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F17: Functional limit theorems; invariance principles 60G52: Stable processes 60G55: Point processes 92D15: Problems related to evolution

beta coalescent evolving coalescent total branch length total external length number of mergers stable moving average processes

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Kersting, Götz; Schweinsberg, Jason; Wakolbinger, Anton. The evolving beta coalescent. Electron. J. Probab. 19 (2014), paper no. 64, 27 pp. doi:10.1214/EJP.v19-3332. https://projecteuclid.org/euclid.ejp/1465065706

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