Weak convergence to the coalescent in neutral population models. (English) Zbl 0938.92024

Möhle, M.

J. Appl. Probab. 36, No. 2, 446-460 (1999).

For a large class of discrete time models the ancestry of a sample of \(n\) individuals (or gens) from the current population having total size \(N\) can be traced back, under suitable time-scaling, when \(N\to\infty\); see, e.g., the author’s paper, ibid. 35, No. 2, 438-447 (1998; Zbl 0913.60022), and references therein. Having established in that work the convergence of finite-dimensional distributions of the ancestral process to those of the n-coalescent (i.e., a certain-continuous time Markov chain), the author shows that under the same assumptions the weak convergence in \(D_E([0,\infty))\) occurs. Here \(E\) is the set of all equivalence relations on \(\{1,\ldots,n\}\) and \(D_E([0,\infty))\) is the set of all E-valued cadlag functions on \([0,\infty)\).
This result, besides the classical exchangeable models, covers models with deterministic variable populations and possibly non-exchangeable offspring variables. It is used to approximate hitting probabilities of the time-scaled ancestral process. A certain type of mutation process can also be imposed upon the genealogical process.

Reviewer: Alexander V.Bulinskij (Moskva)

MSC:

92D10	Genetics and epigenetics
60F05	Central limit and other weak theorems
60G35	Signal detection and filtering (aspects of stochastic processes)

Keywords:

coalescence probability; hitting probability; jump chains; neutrality; genealogical processes; population genetics; robustness; weak convergence

Citations:

Zbl 0913.60022