Sampling probabilities, diffusions, ancestral graphs, and duality under strong selection

Wright–Fisher diffusions and their dual ancestral graphs occupy a central role in the study of allele frequency change and genealogical structure, and they provide expressions, explicit in some special cases but generally implicit, for the sampling probability, a crucial quantity in inference. Under a finite-allele mutation model, with possibly parent-dependent mutation, we consider the asymptotic regime where the selective advantage of one allele grows to infinity, while the other parameters remain fixed. In this regime, we show that the Wright–Fisher diffusion can be approximated either by a Gaussian process or by a process whose components are independent continuous-state branching processes with immigration, aligning with analogous results for Wright–Fisher models but employing different methods. While the first process becomes degenerate at stationarity, the latter does not and provides a simple, analytic approximation for the leading term of the sampling probability. Furthermore, using another approach based on a recursion formula, we characterise all remaining terms to provide a full asymptotic expansion for the sampling probability. Finally, we study the asymptotic behaviour of the rates of the block-counting process of the conditional ancestral selection graph and establish an asymptotic duality relationship between this and the diffusion.