  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.
