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In population genetic studies, the allele frequency spectrum (AFS)efficiently summarizes genome-wide polymorphism data and shapes a variety ofallele frequency-based summary statistics. While existing theory typicallyfeatures equilibrium conditions, emerging methodology requires an analyticalunderstanding of the build-up of the allele frequencies over time. In thiswork, we use the framework of Poisson random fields to derive newrepresentations of the non-equilibrium AFS for the case of a Wright-Fisherpopulation model with selection. In our approach, the AFS is a scaling-limit ofthe expectation of a Poisson stochastic integral and the representation of thenon-equilibrium AFS arises in terms of a fixation time probabilitydistribution. The known duality between the Wright-Fisher diffusion process anda birth and death process generalizing Kingman's coalescent yields anadditional representation. The results carry over to the setting of a randomsample drawn from the population and provide the non-equilibrium behavior ofsample statistics. Our findings are consistent with and extend a previousapproach where the non-equilibrium AFS solves a partial differential forwardequation with a non-traditional boundary condition. Moreover, we provide abridge to previous coalescent-based work, and hence tie several frameworkstogether. Since frequency-based summary statistics are widely used inpopulation genetics, for example, to identify candidate loci of adaptiveevolution, to infer the demographic history of a population, or to improve ourunderstanding of the underlying mechanics of speciation events, the presentedresults are potentially useful for a broad range of topics.

The non-equilibrium allele frequency spectrum in a Poisson random field framework