Absorption time and tree length of the Kingman coalescent and the Gumbel distribution

Martin Möhle; Helmut Pitters

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arXiv · 1502.03983

Absorption time and tree length of the Kingman coalescent and the Gumbel distribution

Martin Möhle·Helmut Pitters

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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.

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Martin Möhle, Helmut Pitters. 2015-02-13. Absorption time and tree length of the Kingman coalescent and the Gumbel distribution. https://arxiv.org/abs/1502.03983

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