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

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 jth cumulant and the jth moment of T are linear combinations of the values ζ(2m), m ∈ {0, . . . , bj/2c}, of the Riemann zeta function ζ 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 Ln 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.