On the external branches of coalescents with multiple collisions. (English) Zbl 1285.60079

Dhersin, Jean-Stéphane; Möhle, Martin

Electron. J. Probab. 18, Paper No. 40, 11 p. (2013).

Summary: A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (lambda-coalescents) is provided. This recursion is used to derive asymptotic results as the sample size \(n\) tends to infinity for the joint moments of the external branch lengths and for the moments of the total external branch length of the Bolthausen-Sznitman coalescent. These asymptotic results are based on a differential equation approach, which is as well useful to obtain exact solutions for the joint moments of the external branch lengths for the Bolthausen-Sznitman coalescent. The results for example show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen-Sznitman coalescent, whereas they are negatively correlated for the Kingman coalescent provided that \(n \geq 4\).

MSC:

60J25	Continuous-time Markov processes on general state spaces
34E05	Asymptotic expansions of solutions to ordinary differential equations
60C05	Combinatorial probability
60J85	Applications of branching processes
92D15	Problems related to evolution
92D25	Population dynamics (general)

Keywords:

asymptotic expansions; Bolthausen-Sznitman coalescent; external branches; joint moments; Kingman coalescent; multiple collisions