Journal of Applied Probability

Convergence to the structured coalescent process

Ryouta Kozakai, Akinobu Shimizu, and Morihiro Notohara

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Abstract

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

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

Dates
First available in Project Euclid: 17 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1466172870

Mathematical Reviews number (MathSciNet)
MR3514294

Zentralblatt MATH identifier
1342.92181

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

Keywords
Structured coalescent nonconservative migration Cannings' reproduction weak convergence

Citation

Kozakai, Ryouta; Shimizu, Akinobu; Notohara, Morihiro. Convergence to the structured coalescent process. J. Appl. Probab. 53 (2016), no. 2, 502--517. https://projecteuclid.org/euclid.jap/1466172870


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