2026/03/12 by Janique Krasnowska, Paul Jenkins, Adam Johansen · 1 voice
Mathematics · #math.PR
Consider a population evolving as a discrete-time supercritical multi-type Galton--Watson process. Suppose we run the process for T generations, then sample k individuals uniformly at generation T and trace their genealogy backwards in time. In the limiting regime as T → ∞, the expected behaviour of the sample's ancestry has been analysed extensively in the single-type case and, more recently, for multi-type processes in the critical case. In this paper, we present a formula for the distribution function of the generation t of the most recent common ancestor in terms of the limiting distribution of the normalised population size. In addition, we provide effective bounds for the decay of this distribution function to 1 in terms of the harmonic moments of the population size at generation t. In order to better understand the behaviour of these harmonic moments, we use a multi-type generalisation of the Harris--Sevastyanov transformation to express harmonic moments at generation t in terms of moments of the transformed process at the first generation. We present numerical results demonstrating that it is possible to approximate the coalescence time distribution effectively in practical settings.