The coalescent structure of multitype continuous-time Galton-Watson trees

TL;DR

This work analyzes the genealogy of a uniformly chosen k‑particle sample from a large population evolving according to a critical continuous‑time multitype Galton–Watson process with finite second moments. The authors develop a spine framework and a two‑step change of measure that yields a tractable description of the k spines as a size‑biased, uniformly sampled subset whose descendants follow a Markov branching structure. They prove a universal coalescent limit: after a deterministic time‑change, the sampled genealogy has Kingman‑type topology when types are ignored, while the coalescent times retain a multitype signature through explicit type‑dependent weights; in the limit, only binary splits occur, with the detailed joint law encoded by a product of type‑weights and a universal integral term. The results generalize spine techniques from the single‑type setting to the multitype case, revealing how inter‑type interactions shape the ancestral structure and providing precise asymptotics for splitting times, partition processes, and sampling under the original measure. This advances understanding of multitype genealogies and yields connections to classical coalescent theory in a rich, type‑structured environment.

Abstract

We investigate the genealogy of a sample of $k\geq1$ particles chosen uniformly without replacement from a population alive at large times in a critical continuous-time multitype Galton-Watson process with finite second moment. We will show that subject to a deterministic time-change, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman's coalescent, when the types are discarded, and the coalescent times of the $k-1$ pairwise mergers look like a mixture of independent identically distributed times. We show that such an ancestral lineage in the limit, strongly depends on the multitype offspring distribution, which differs from the single type case Harris, Johnston, and Roberts [Annals of Applied Probability, 2020]. Our approach uses $k$ distinguished 'spine' particles and a suitable change of measure under which (a) the spines form a uniform sample without replacement that depend on the colours but additionally (b) there is $k$-size biasing and discounting according to the population size. Our work substantially extends the spine techniques developed in Harris, Johnston, and Roberts [Annals of Applied Probability, 2020] for genealogies of uniform samples of size $k$ in critical, continuous-time, single-type Galton-Watson processes. We generalize these methods to the multi-type setting and provide a comprehensive analysis of how functionals of the spines are influenced by the types. While the single-type case exhibits a more homogeneous structure with simpler dependency patterns, the multi-type case introduces interactions between different types, leading to a more intricate dependency structure where functionals must account for type-specific behaviours and inter-type relationships.

Figures (3)

• Figure 1: 2-type MBGW tree under $\mathbb{P}^{(8)}_1$, where individuals type 1 are depicted with a continuous line, and individuals type 2 are depicted with a dashed line. The time of death of a particle is represented by a dot. The births-off the spine are represented with a dotted line.
• Figure 2: We show a small window of a multitype tree, where there is a splitting of the spines at time $t_h$, and we follow individual $v(h,1,3)$, which is the third individual type 1. This individual carries $k_{v(h,1,3)}$ spines, and the spines split at time $t_{v(h,1,3)}$ with type $c(v(h,1,3))$. Observe that the time $t_{v(h,1,3)}$ and the type $c(v(h,1,3))$ correspond to one of the remaining $\tau_{j}$ and $i_j$, for $j>h$.
• Figure 3: Illustration of the vertices carrying marks, in a 3 type tree, starting from 8 marks. The type 1 individuals are depicted in black, type 2 in red and type 3 in blue. At the first spine splitting event, occurring at time $t_1-$, the vertex $v(1)$ type $i_1=1$, has ${\bf L } _{v(1)}$ children, and from them ${\bf g } _1=(2,1,2)$ carry at least one mark. The partition ${\bf P } _1$ consists of $P_{1,1}=\{A_{1,1,1},A_{1,1,2}\}$, $P_{1,2}=\{A_{1,2,1}\}$ and $P_{1,3}=\{A_{1,3,1},A_{1,3,2}\}$. Only two blocks will split at future times, namely $A_{1,1,2}$ and $A_{1,3,1}$. At time $t^{(1,2)}_{1}-$ the individual $v(1,1,2)$, which has type $i^{(1,2)}_1=1$, undergoes a spine splitting event, having $\bm \ell^{(1,2)}_1$ offspring, and ${\bf g } ^{(1,2)}_1=(1,1,0)$ carry at least one mark. The partition that it generates is ${\bf P } ^{(1,2)}_1$ consisting of blocks $P^{(1,2)}_{1,1}=\{A^{(1,2)}_{1,1,1}\}$ and $P^{(1,2)}_{1,2}=\{A^{(1,2)}_{1,2,1}\}$. For block $A_{1,1,2}$ there are $n^{(1,2)}=2$ spine splitting events after $t_1$. The second one, occurs at time $t^{(1,2)}_2$, from an individual type $i^{(1,2)}_2=3$, giving rise to a partition ${\bf P } ^{(1,2)}_2$ with $P^{(1,2)}_{2,2}=\{A^{(1,2)}_{2,2,1},A^{(1,2)}_{2,2,2}\}$.

Theorems & Definitions (48)

• Proposition 1
• Theorem 1
• Lemma 1
• Lemma 2: Going back to the $\mathbb{P}^{(k)}_{unif,T,r}$ measure
• proof
• Lemma 3: Markov branching property
• proof
• Lemma 4
• proof
• Lemma 5