  In mathematical population genetics, it is well known that one can represent
the genealogy of a population by a tree, which indicates how the ancestral
lines of individuals in the population coalesce as they are traced back in
time. As the population evolves over time, the tree that represents the
genealogy of the population also changes, leading to a tree-valued stochastic
process known as the evolving coalescent. Here we will consider the evolving
coalescent for populations whose genealogy can be described by a beta
coalescent, which is known to give the genealogy of populations with very large
family sizes. We show that as the size of the population tends to infinity, the
evolution of certain functionals of the beta coalescent, such as the total
number of mergers, the total branch length, and the total length of external
branches, converges to a stationary stable process. Our methods also lead to
new proofs of known asymptotic results for certain functionals of the
non-evolving beta coalescent.
