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2014 The evolving beta coalescent
Götz Kersting, Jason Schweinsberg, Anton Wakolbinger
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Electron. J. Probab. 19: 1-27 (2014). DOI: 10.1214/EJP.v19-3332

Abstract

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.

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Götz Kersting. Jason Schweinsberg. Anton Wakolbinger. "The evolving beta coalescent." Electron. J. Probab. 19 1 - 27, 2014. https://doi.org/10.1214/EJP.v19-3332

Information

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

zbMATH: 1302.60137
MathSciNet: MR3238784
Digital Object Identifier: 10.1214/EJP.v19-3332

Subjects:
Primary: 60K35
Secondary: 60F17 , 60G52 , 60G55 , 92D15

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

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