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Convergence to the structured coalescent process

Part of: Limit theorems

Published online by Cambridge University Press:  21 June 2016

Ryouta Kozakai ,
Akinobu Shimizu  and
Morihiro Notohara
Ryouta Kozakai*
Affiliation:
Nagoya City University
Akinobu Shimizu*
Affiliation:
Nagoya City University
Morihiro Notohara*
Affiliation:
Nagoya City University
*
* Postal address: Graduate School of Natural Sciences, Nagoya City University, Mizuho, Nagoya, 467-8501, Japan.
* Postal address: Graduate School of Natural Sciences, Nagoya City University, Mizuho, Nagoya, 467-8501, Japan.
* Postal address: Graduate School of Natural Sciences, Nagoya City University, Mizuho, Nagoya, 467-8501, Japan.
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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.

Keywords

Structured coalescentnonconservative migrationCannings' reproductionweak convergence

MSC classification

Primary: 92D25: Population dynamics (general) 92D10: Genetics
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 502 - 517
Copyright
Copyright © Applied Probability Trust 2016 

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