A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing. (English) Zbl 0910.60007

Möhle, M.

Adv. Appl. Probab. 30, No. 2, 493-512 (1998).

Summary: A simple convergence theorem for sequences of Markov chains is presented in order to derive new ‘convergence-to-the-coalescent’ results for diploid neutral population models. For the so-called diploid Wright-Fisher model with selfing probability \(s\) and mutation rate \(\theta\), it is shown that the ancestral structure of \(n\) sampled genes can be treated in the framework of an \(n\)-coalescent with mutation rate \(\widetilde \theta:= \theta(1-s/2)\), if the population size \(N\) is large and if the time is measured in units of \((2-s)N\) generations.

MSC:

60F05	Central limit and other weak theorems
92D10	Genetics and epigenetics
92D25	Population dynamics (general)
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

coalescent; diploid population models; genealogical process; population genetics; robustness; selfing