Robustness results for the coalescent. (English) Zbl 0913.60022

Möhle, M.

J. Appl. Probab. 35, No. 2, 438-447 (1998).

This paper is concerned with a convergence theorem to the coalescent in a genealogical process. The usual conditions required for such theorems to hold rely on assumptions of exchangeability; the author dispenses with these in his paper. By an analysis of the coalescence probability, i.e., the probability that two genes chosen at random without replacement from a gene population of size \(N\) have a common ancestor in the previous generation, the author is able to prove his main theorem. This states that for a large class of non-exchangeable population models, there is convergence of the finite-dimensional distributions of a certain well defined process \(\{R_{\tau_N}(t)\}_{t\in T}\) to those of the \(n\)-coalescent, as the number \(N\) of genes in the population tends to infinity. This generalizes the earlier proof of J. F. C. Kingman [in: Exchangeability in probability and statistics, 97-112 (1982; Zbl 0494.92011)] for a class of exchangeable models.

Reviewer: J.M.Gani (Canberra)

MSC:

60F05	Central limit and other weak theorems
92D10	Genetics and epigenetics
60J27	Continuous-time Markov processes on discrete state spaces
92D25	Population dynamics (general)

Keywords:

coalescence probability; genealogical process; population genetics; Moran and Fisher-Wright models

Citations:

Zbl 0494.92011