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On the genealogy of multi-type Cannings models and their limiting exchangeable coalescents

Maximilian Flamm, Martin Möhle

TL;DR

The paper develops a rigorous foundation for multi-type genealogies under Cannings-type reproduction by formulating a multi-type ancestral process on labeled partitions and proving the existence of limiting multi-type exchangeable coalescents. It introduces diagonal-tensor rate representations and integral representations on the Delta-j simplices, enabling a generalization of $\Lambda$- and $\Xi$-coalescents to multi-type populations. The authors establish discrete-time and continuous-time limiting regimes, with concrete illustrations via a multi-type Wright–Fisher model that yields a multi-type Kingman coalescent under calibration, and a mutation-only model with no coalescence. These results extend coalescent theory to multi-type settings, providing a framework for analyzing type-structured genealogies and their limiting dynamics in large populations.

Abstract

We study the multi-type Cannings population model. Each individual has a type belonging to a given at most countable type space $E$. The population is hence divided into $|E|$ subpopulations. The subpopulation sizes are assumed to be constant over the generations, whereas the number of offspring of type $\ell\in E$ of all individuals of type $k\in E$ is allowed to be random. Under a joint exchangeability assumption on the offspring numbers, the transition probabilities of the ancestral process of a sample of individuals satisfy a multi-type consistency property, paving a way to prove in the limit for large subpopulation sizes the existence of multi-type exchangeable coalescent processes via Kolmogorov's extension theorem. Integral representations for the infinitesimal rates of these multi-type exchangeable coalescents and some of their properties are studied. Examples are provided, among them multi-type Wright-Fisher models and multi-type pure mutation models. The results contribute to the foundations of multi-type coalescent theory and provide new insights into (the existence of) multi-type exchangeable coalescents.

On the genealogy of multi-type Cannings models and their limiting exchangeable coalescents

TL;DR

The paper develops a rigorous foundation for multi-type genealogies under Cannings-type reproduction by formulating a multi-type ancestral process on labeled partitions and proving the existence of limiting multi-type exchangeable coalescents. It introduces diagonal-tensor rate representations and integral representations on the Delta-j simplices, enabling a generalization of - and -coalescents to multi-type populations. The authors establish discrete-time and continuous-time limiting regimes, with concrete illustrations via a multi-type Wright–Fisher model that yields a multi-type Kingman coalescent under calibration, and a mutation-only model with no coalescence. These results extend coalescent theory to multi-type settings, providing a framework for analyzing type-structured genealogies and their limiting dynamics in large populations.

Abstract

We study the multi-type Cannings population model. Each individual has a type belonging to a given at most countable type space . The population is hence divided into subpopulations. The subpopulation sizes are assumed to be constant over the generations, whereas the number of offspring of type of all individuals of type is allowed to be random. Under a joint exchangeability assumption on the offspring numbers, the transition probabilities of the ancestral process of a sample of individuals satisfy a multi-type consistency property, paving a way to prove in the limit for large subpopulation sizes the existence of multi-type exchangeable coalescent processes via Kolmogorov's extension theorem. Integral representations for the infinitesimal rates of these multi-type exchangeable coalescents and some of their properties are studied. Examples are provided, among them multi-type Wright-Fisher models and multi-type pure mutation models. The results contribute to the foundations of multi-type coalescent theory and provide new insights into (the existence of) multi-type exchangeable coalescents.

Paper Structure

This paper contains 18 sections, 13 theorems, 122 equations, 3 figures.

Table of Contents

  1. Introduction and model definition
  2. Ancestral structure
  3. Block labeled set partitions
  4. Random labeled partitions
  5. Multi-type ancestral process
  6. Consistency and symmetry
  7. Limiting results
  8. Multi-type exchangeable coalescents
  9. Integral representations
  10. Examples
  11. Multi-type Wright--Fisher model
  12. A multi-type mutation model without merging events
  13. Proofs
  14. Proof of Theorem \ref{['main3']}
  15. Appendix
  16. ...and 3 more sections

Key Result

Proposition 1

Under (A), the functions $\Phi_j:{\cal T}_j\to[0,1]$, $j:=(j_k)_{k\in E}\in{\mathbb N}_0^E$, are consistent in the following sense. For all $j=(j_k)_{k\in E}\in{\mathbb N}_0^E$ and all tensors $T\in{\cal T}_j$, the equality holds for each $\ell\in E$ with $i_\ell:=\sum_{k\in E}\sum_{s\in[j_k]}i_{k,\ell,s}<N_\ell$, where $e_k$ denotes the $k$-th unit vector in ${\mathbb R}^E$, the tensor $T(k,\ell

Figures (3)

  • Figure 1: An illustration of two consecutive generations with three subpopulations
  • Figure 2: A transition step of the limiting process $\Pi$ for $E=\{1,2,3\}$ and $j=(4,5,7)$ corresponding to the tensor $T$ in (\ref{['tensorexample']})
  • Figure 3: Transition matrix $P$ of the ancestral process for sample size $n=2$ and $E=\{1,2\}$

Theorems & Definitions (30)

  • Definition 1: Partial exchangeability
  • Proposition 1: Multi-type consistency
  • Corollary 1: Monotonicity
  • Corollary 2: Natural coupling
  • Lemma 1: Exchangeability
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Proposition 2
  • Theorem 3
  • ...and 20 more