Uniform sampling of multitype continuous-time Bienaym\'e-Galton-Watson trees

TL;DR

This paper develops a general framework for uniform sampling of genealogies in multitype continuous-time Bienaymé--Galton--Watson processes, revealing complex interactions between types and providing explicit genealogical descriptions.

Contribution

It introduces a $k$-spine decomposition and change of measure approach for multitype processes, extending single-type results and enabling detailed genealogical analysis.

Findings

Characterizes spine splitting times and offspring distributions.

Provides an explicit description of ancestral structures across types.

Reveals rich interactions between different types in the genealogy.

Abstract

We study the genealogy of a sample of $k$ individuals taken uniformly without replacement from a continuous-time multitype Bienaym\'e--Galton--Watson process at fixed times. Our results are quite general, requiring only that the process be non-simple and conservative, and that every type has a positive probability to ``eventually lead to'' all other types within the population. The corresponding single-type case has recently been studied by Johnston (2019), Harris, Johnston, and Roberts (2020), and Harris, Johnston, and Pardo (2024). Our approach is based on a $k$-spine decomposition and a suitable change of measure under which the distinguished spines form a uniform sample at time $T$, while the population size is subject to $k$-size biasing and exponential discounting. This construction preserves a branching Markov property and yields an explicit description of the genealogical tree at fixed times. In particular, we characterise spine splitting times, offspring distributions, and type-dependent ancestral structures, revealing rich interactions between types that are absent in the single-type setting. The present results form the basis of a forthcoming series of papers in which limiting genealogical behaviour is analysed under various asymptotic regimes and more general sampling schemes by the authors, see Angtuncio et al. (2026b), (2026c) and (2026d).