  The site frequency spectrum (SFS) is a widely used summary statistic of
genomic data. Motivated by recent evidence for the role of neutral evolution in
cancer, we investigate the SFS of neutral mutations in an exponentially growing
population. Using branching process techniques, we establish (first-order)
almost sure convergence results for the SFS of a Galton-Watson process,
evaluated either at a fixed time or at the stochastic time at which the
population first reaches a certain size. We finally use our results to
construct consistent estimators for the extinction probability and the
effective mutation rate of a birth-death process.
