Some uses of ordered representations in finite-population exchangeable ancestry models
For a population model that encodes parent-child relations, an ordered representation is a partial or complete labelling of individuals, in order of their descendants’ long-term success in some sense, with respect to which the ancestral structure is more tractable. The two most common types are the lookdown and the spinal decomposition(s), used respectively to study exchangeable models and Markov branching processes. We study the lookdown for an exchangeable model with a fixed, arbitrary sequence of natural numbers, describing population size over time. We give a simple and intuitive construction of the lookdown via the complementary notions of forward and backward neutrality. We discuss its connection to the spinal decomposition in the setting of Galton-Watson trees. We then use the lookdown to give sufficient conditions on the population sequence for the existence of a unique infinite line of descent. For a related but slightly weaker property, takeover, the necessary and sufficient conditions are more easily expressed: infinite time passes on the coalescent time scale. The latter property is also related to the following question of identifiability: under what conditions can some or all of the lookdown labelling be determined by the unlabelled lineages? A reasonably good answer can be obtained by comparing extinction times and relative sizes of lineages.