Asymptotics for the site frequency spectrum associated with the genealogy of a birth and death process

Consider a birth and death process started from one individual in which each individual gives birth at rate λ and dies at rate μ, so that the population size grows at rate r=λ−μ. Lambert (Theor. Popul. Biol. 122 (2018) 30–35) and Harris, Johnston, and Roberts (Ann. Appl. Probab. 30 (2020) 1368–1414) came up with methods for constructing the exact genealogy of a sample of size n taken from this population at time T. We use the construction of Lambert, which is based on the coalescent point process, to obtain asymptotic results for the site frequency spectrum associated with this sample. In the supercritical case r>0, our results extend results of Durrett (Ann. Appl. Probab. 23 (2013) 230–250) for exponentially growing populations. In the critical case r=0, our results parallel those that Dahmer and Kersting (Ann. Appl. Probab. 25 (2015) 1325–1348) obtained for Kingman’s coalescent.