On the block counting process and the fixation line of exchangeable coalescents. (English) Zbl 1346.60124

Gaiser, Florian; Möhle, Martin

ALEA, Lat. Am. J. Probab. Math. Stat. 13, No. 2, 809-833 (2016).

Summary: We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process is Siegmund dual to the fixation line. For exchangeable coalescents restricted to a sample of size \(n\) and with dust we provide a convergence result for the block counting process as \(n\) tends to infinity. The associated limiting process is related to the frequencies of singletons of the coalescent. Via duality we obtain an analogous convergence result for the fixation line of exchangeable coalescents with dust. The Dirichlet coalescent and the Poisson-Dirichlet coalescent are studied in detail.

MSC:

60J27	Continuous-time Markov processes on discrete state spaces
60G09	Exchangeability for stochastic processes
60F05	Central limit and other weak theorems
92D10	Genetics and epigenetics

Keywords:

exchangeable coalescent; block counting process; duality; fixation line; Dirichlet coalescent; Poisson-Dirichlet coalescent