On the Coalescence Time Distribution in Multi-type Supercritical Branching Processes

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 $T \rightarrow \infty$, 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.

💡 Research Summary

The paper studies the genealogy of a supercritical multi‑type Galton–Watson branching process observed at a large time T. After running the process for T generations, k ≥ 2 individuals are sampled uniformly without replacement from generation T and the generation t of their most recent common ancestor (MRCA) is examined. The authors extend previous results that were limited to either single‑type processes or multi‑type processes with zero extinction probability and a finite number of types.

First, the authors recall the standard framework: a countable type space S, an irreducible mean offspring matrix M, and the supercritical regime characterized by a convergence radius R ∈ (0,1). Under these assumptions there exist a positive left eigenvector ν and a right eigenvector u (normalized so that u·ν = 1 and ν·1 = 1) such that the normalized population size R^T|Z_T| converges in L² to a random limit W(i₀) that depends on the type i₀ of the founding ancestor. This classical result (Theorem 1) provides the foundation for the later analysis.

The main contribution is Theorem 3, which gives an exact expression for the limiting probability that the MRCA lies before generation t, conditional on the population at generation T being at least k. The formula involves the random variables W(i) for all types i∈S and reads

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