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The Yule-$Λ$ Nested Coalescent: Distribution of the Number of Lineages

Toni Gui

TL;DR

The paper analyzes the Yule-$\Lambda$ nested coalescent, combining a Yule species-branching process with within-species $\Lambda$-coalescence. It introduces the fixed-point operator $G_c$ and proves the existence of a finite-mean fixed point $\nu_c^*$ in $\mathcal{S}_1$ when $0<c<E[1/X]$, along with convergence in distribution of the counts of lineages per surviving species to $\nu_c^*$. It further shows that for some $c$ there exists an additional fixed point $\nu_c'$ in $\mathcal{S}$ which may have infinite mean, with the mass at infinity depending on $P(X=1)$. The results rely on recursive distributional equations, generating function techniques, and delicate probabilistic couplings, highlighting both uniqueness in the finite-mean regime and non-uniqueness when infinite mean is possible. These findings contribute to understanding multi-species genealogies under Λ-coalescence, revealing how inter-species dynamics affect the distribution of intra-species lineages and the role of dust in fixed-point behavior.

Abstract

We study a model of a population with individuals sampled from different species. The Yule-$Λ$ nested coalescent describes the genealogy of the sample when each species merges with another randomly chosen species with a constant rate $c$ and the mergers of individuals in each species follow the $Λ$-coalescent. For the Yule-$Λ$ nested coalescent with $c<\int_0^1x^{-1}Λ(dx)<\infty$, where $Λ$ is the measure that characterizes the $Λ$-coalescent, we show that under some initial conditions, the distribution of the number of individual lineages belonging to one species converges weakly to the distribution $ν_c^*$, which is the solution to some recursive distributional equation (RDE) with finite mean. In addition, we show that for some values of $c$, the RDE has another solution with infinite mean.

The Yule-$Λ$ Nested Coalescent: Distribution of the Number of Lineages

TL;DR

The paper analyzes the Yule- nested coalescent, combining a Yule species-branching process with within-species -coalescence. It introduces the fixed-point operator and proves the existence of a finite-mean fixed point in when , along with convergence in distribution of the counts of lineages per surviving species to . It further shows that for some there exists an additional fixed point in which may have infinite mean, with the mass at infinity depending on . The results rely on recursive distributional equations, generating function techniques, and delicate probabilistic couplings, highlighting both uniqueness in the finite-mean regime and non-uniqueness when infinite mean is possible. These findings contribute to understanding multi-species genealogies under Λ-coalescence, revealing how inter-species dynamics affect the distribution of intra-species lineages and the role of dust in fixed-point behavior.

Abstract

We study a model of a population with individuals sampled from different species. The Yule- nested coalescent describes the genealogy of the sample when each species merges with another randomly chosen species with a constant rate and the mergers of individuals in each species follow the -coalescent. For the Yule- nested coalescent with , where is the measure that characterizes the -coalescent, we show that under some initial conditions, the distribution of the number of individual lineages belonging to one species converges weakly to the distribution , which is the solution to some recursive distributional equation (RDE) with finite mean. In addition, we show that for some values of , the RDE has another solution with infinite mean.
Paper Structure (12 sections, 16 theorems, 351 equations, 2 figures)

This paper contains 12 sections, 16 theorems, 351 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The $\Lambda$-coalescent
  3. Nested Coalescents
  4. Definition of the Yule-$\Lambda$ Nested Coalescent Model
  5. Main Results for the Yule-$\Lambda$ Nested Coalescent
  6. Existence and Uniqueness of the Solution on $\mathcal{S}_1$ to the RDE (\ref{['theoremequ1']})
  7. Existence of the Solution to the RDE
  8. Uniqueness of the Solution to the RDE
  9. Proof of Theorem \ref{['theo4unilambda']}
  10. Proof of Theorem \ref{['Theoexistinflambda']}
  11. Existence of the Solution on $\mathcal{S}$ with Infinite Mean to the RDE (\ref{['theoremequ1']})
  12. Proof of Theorem \ref{['Theoconvlambda1']}

Key Result

Theorem 1

If $c<E[1/X]<\infty$, then there exists a unique distribution $\nu_c^*$ in $\mathcal{S}_1$ such that

Figures (2)

  • Figure 1: (From YuleKingmanToni) A nested coalescent tree which illustrates a model starting from 4 species and 3, 4, 2, and 3 individual lineages in species 1, 2, 3, and 4 respectively. The dark lines present a possible ancestral tree for the sampled individual lineages and the light blue tree in the background shows a possible species tree. Only individual lineages belonging to the same species can merge.
  • Figure 2: Two binary species trees representing Model 1 and Model 2.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 22 more