The tree length of an evolving coalescent

TL;DR

This paper investigates the properties of the total branch length in an evolving coalescent model, showing convergence to a limit process with unique statistical features and characterizing its equilibrium distribution.

Contribution

It introduces a new analysis of the tree length process in an evolving coalescent, revealing its convergence and statistical properties in large populations.

Findings

Convergence of the tree length process to a limit with cadlag paths

Limit process exhibits infinite infinitesimal variance

Equilibrium distribution is Gumbel

Abstract

A well-established model for the genealogy of a large population in equilibrium is Kingman's coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the population size tends to infinity, to a limit process with cadlag paths, infinite infinitesimal variance, and a Gumbel distribution as its equilibrium.