Run a supercritical branching process for T generations. Sample k individuals from generation T. Trace their genealogy backward. The most recent common ancestor lived at some generation t — how is t distributed?
For single-type processes, this is well understood. For multi-type processes — where individuals have types that affect their offspring distributions — the coalescence time distribution has been analyzed only in the critical case. The supercritical multi-type case yields a formula expressing the distribution function of the MRCA generation in terms of the limiting distribution of the normalized population size (arXiv:2603.11990).
The decay of this distribution — how quickly the probability that the MRCA is at generation t approaches certainty as t grows — is bounded using harmonic moments of the population size. Harmonic moments (expectations of negative powers of population size) are harder to control than ordinary moments because they are sensitive to the probability of the population being small. In a supercritical process, the population grows exponentially on average, but individual realizations can be small by chance, and these rare events dominate the harmonic moments.
The Harris-Sevastyanov transformation converts the multi-type harmonic moment problem into a moment problem for a transformed process at the first generation. The transformation changes the measure so that what was a harmonic moment becomes an ordinary moment — tractable by standard tools.
The practical message: coalescence time in structured populations is computable, not just asymptotically but for finite populations with multiple types. The genealogy of a sample from a complex population is no more mysterious than the branching process that generated it — once the right transformation is applied.