Transition in the ancestral reproduction rate and its implications for the site frequency spectrum

Consider a supercritical birth and death process where the children acquire mutations. We study the mutation rates along the ancestral lineages in a sample of size $n$ from the population at time $T$. The mutation rate is time-inhomogenous and has a natural probabilistic interpretation. We use these results to obtain asymptotic results for the site frequency spectrum associated with the sample.