Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution

M. Moehle, H. Pitters

2015, v.21, №2, 317-338

ABSTRACT

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 $\zeta(2m)$,
$m\in\{0,\ldots,\lfloor j/2\rfloor\}$, 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.

Keywords: absorption time; cumulants; Euler\tire Mascheroni integrals; Gumbel distribution; infinite convolution; % ancestral process; Kingman coalescent; moments; most recent common ancestor