Absorption time and tree length of the Kingman coalescent and the Gumbel distribution. (English) Zbl 1328.60021

Möhle, M.; Pitters, H.

Markov Process. Relat. Fields 21, No. 2, 317-338 (2015).

Summary: Formulas are provided for the cumulants and the moments of the time \(T\) back to the most recent common ancestor of the Kingman coalescent. It is shown that both the \(j\)th cumulant and the \(j\)th moment of \(T\) are linear combinations of the values \(\zeta(2m)\), \(m\in\{0,\dots,\lfloor j/2\rfloor\}\), of the Riemann zeta function \(\zeta\) with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length \(L_n\) of the Kingman coalescent restricted to a sample of size \(n\) is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.

MSC:

60C05	Combinatorial probability
60J28	Applications of continuous-time Markov processes on discrete state spaces
60G50	Sums of independent random variables; random walks
92D25	Population dynamics (general)

Keywords:

absorption time; cumulants; Euler-Mascheroni integrals; Gumbel distribution; infinite convolution; Kingman coalescent; moments; most recent common ancestor; tree length; zeta function