The coalescent structure of Galton-Watson trees in varying environments

Harris Simon C., Palau Sandra, Pardo Juan Carlos

Published 2022-07-22Version 1

We investigate the genealogy of a sample of k particles chosen uniformly without replacement from a population alive at a large time n in a critical discrete-time Galton-Watson process in a varying environment (GWVE). We will show that subject to an explicit deterministic time-change involving only the mean and variances of the varying offspring distributions, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman's coalescent, and the coalescent times of the k-1 pairwise mergers look like a mixture of independent identically distributed times. Our approach uses k spine particles and a change of measure $\mathbf{Q}_n^{(e,k, \theta)}$ under which the spines form a uniform sample without replacement at time n, as required, but additionally there is k-size biasing and discounting at rate $\theta$ by the population size at time n. We give a forward in time construction for the process with k spines under $\mathbf{Q}_n^{(e,k, \theta)}$ where the k size-biasing provides a great deal of structural independence that enables explicit calculations, and the discounting means that no additional offspring moment assumptions are required. Combining the special properties of $\mathbf{Q}_n^{(e,k, \theta)}$ together with the Yaglom theorem for GWVE, plus a suitable time-change that encodes variation within the environment, we then undo the k-size biasing and discounting to recover the limiting genealogy for a uniform sample of size k in the GWVE. Our work extends the spine techniques developed in Harris, et. al [Annals of Applied Probability, 2020] and complements recent work by Kersting [Proceedings of the Steklov Institute of Mathematics, 2022] which describes the genealogy of the entire extant population in the GWVE in the large time limit