Site Frequency Spectrum in stationary branching populations

This paper explores the Site Frequency Spectrum (SFS) in stationary branching populations. We derive estimates for the SFS associated with a sample from a continuous-state branching process conditioned to never go extinct, utilizing a quadratic branching mechanism. The genealogy of such processes is represented by a real tree with a semi-infinite branch, and we compute the expectation of the SFS under the infinitely-many-sites assumption as the sample size approaches infinity. Additionally, we present a continuum version of the SFS as a random point measure on the positive real line and compute the density of its expected measure explicitly. Finally, we derive estimates for the size of the clonal subpopulation carrying the same genotype as the most recent common ancestor of the whole population at a given time.