Journal of Applied Probability

Convergence to the structured coalescent process

Ryouta Kozakai, Akinobu Shimizu, and Morihiro Notohara

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The coalescent was introduced by Kingman (1982a), (1982b) and Tajima (1983) as a continuous-time Markov chain model describing the genealogical relationship among sampled genes from a panmictic population of a species. The random mating in a population is a strict condition and the genealogical structure of the population has a strong influence on the genetic variability and the evolution of the species. In this paper, starting from a discrete-time Markov chain model, we show the weak convergence to a continuous-time Markov chain, called the structured coalescent model, describing the genealogy of the sampled genes from whole population by means of passing the limit of the population size. Herbots (1997) proved the weak convergence to the structured coalescent on the condition of conservative migration and Wright–Fisher-type reproduction. We will give the proof on the condition of general migration rates and exchangeable reproduction.

Article information

J. Appl. Probab., Volume 53, Number 2 (2016), 502-517.

First available in Project Euclid: 17 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D25: Population dynamics (general) 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 60F17: Functional limit theorems; invariance principles

Structured coalescent nonconservative migration Cannings' reproduction weak convergence


Kozakai, Ryouta; Shimizu, Akinobu; Notohara, Morihiro. Convergence to the structured coalescent process. J. Appl. Probab. 53 (2016), no. 2, 502--517.

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