Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. (English) Zbl 1002.92015

Möhle, M.

Adv. Appl. Probab. 32, No. 4, 983-993 (2000).

Summary: Haploid population models with non-overlapping generations and fixed population size \(N\) are considered. It is assumed that the family sizes \(\nu_1,\dots, \nu_N\) within a generation are exchangeable random variables. Rates of convergence for the finite-dimensional distributions of a properly time-scaled ancestral coalescent process are established and expressed in terms of the transition probabilities of the ancestral process, i.e., in terms of the joint factorial moments of the offspring variables \(\nu_1,\dots, \nu_N\). The Kingman coalescent appears in the limit as the population size \(N\) tends to infinity if and only if triple mergers are asymptotically negligible in comparison with binary mergers. In this case, a simple upper bound for the rate of convergence of the finite-dimensional distributions is derived. It depends (up to some constants) only on the three factorial moments \(E((\nu_1)_2)\), \(E((\nu_1)_3)\) and \(E((\nu_1)_2(\nu_2)_2)\), where \((x)_k:= x(x- 1)\cdots(x- k+1)\). Examples are the Wright-Fisher model, where the rate of convergence is of order \(N^{-1}\), and the Moran model, with a convergence rate of order \(N^{-2}\).

MSC:

92D15	Problems related to evolution
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60G09	Exchangeability for stochastic processes
92D25	Population dynamics (general)

Keywords:

ancestral processes; coalescent; exchangeability; generator; neutrality; population genetics; rate of convergence; total variationa distance; weak convergence