General selection models: Bernstein duality and minimal ancestral structures

Seminar: Probability Seminar | September 18 | 3:10-4 p.m. | 1011 Evans Hall

 Sebastian Hummel, Bielefeld University

 Department of Statistics

We construct a sequence of Moran models that converges for large populations under suitable conditions to the $\Lambda$-Wright-Fisher process with a drift that is vanishing at the boundaries. The genealogical structure inherent in the graphical representation of the finite population models leads in the large population limit to a generalisation of the ancestral selection graph of Krone and Neuhauser. We introduce an ancestral process that keeps track of the sampling distribution along the ancestral structures and that satisfies a duality relation with the type-frequency process. We refer to it as Bernstein coefficient process and to the relation as Bernstein duality. The latter generalizes a recent result by Gonzalez Casanova and Spano, who established for a restricted class of drifts a moment duality between the type-frequency process and the line-counting process of the ancestral selection graph. As an application of the Bernstein duality, I will explain the derivation of criteria for the accessibility of the boundary. This extends results, which were previously only available for the aforementioned class of drifts, to any polynomial drift vanishing at the boundary. An intriguing feature of our construction is that multiple ancestral processes are associated to the same forward dynamics. If time permits, I will explain how to characterize the set of optimal ancestral structures and provide a recipe to construct them from the drift.
This is joint work with F. Cordero and E. Schertzer.