Convergence to the structured coalescent process. (21st June 2016)

• Record Type: Journal Article
• Authors: Kozakai, Ryouta; Shimizu, Akinobu; Notohara, Morihiro
• Is Part Of: Journal of applied probability. Volume 53:Number 2(2016)
• Page Start: 502
• Page End: 517
• Publication Date: 2016-06-21
• Subjects: Structured coalescent, -- nonconservative migration, -- Cannings' reproduction, -- weak convergence; Primary 92D25, -- 92D10, -- Secondary 60F17; 519.2
• DOI: 10.1017/jpr.2016.16
• Languages: English
• ISSNs: 0021-9002

Abstract

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.