Limit theorems for the site frequency spectrum of neutral mutations in an exponentially growing population.

The site frequency spectrum (SFS) is a widely used summary statistic of genomic data, offering a simple means of inferring the evolutionary history of a population. Motivated by recent evidence for the role of neutral evolution in cancer, we examine the SFS of neutral mutations in an exponentially growing population. Whereas recent work has focused on the mean behavior of the SFS in this scenario, here, we investigate the first-order asymptotics of the underlying stochastic process. Using branching process techniques, we show that the SFS of a Galton-Watson process evaluated at a fixed time converges almost surely to a random limit. We also show that the SFS evaluated at the stochastic time at which the population first reaches a certain size converges in probability to a constant. Finally, we illustrate how our results can be used to construct consistent estimators for the extinction probability and the effective mutation rate of a birth-death process.