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

Janique Krasnowska, Paul Jenkins, Adam Johansen

Abstract

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.

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

Abstract

Consider a population evolving as a discrete-time supercritical multi-type Galton--Watson process. Suppose we run the process for generations, then sample individuals uniformly at generation and trace their genealogy backwards in time. In the limiting regime as , 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 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 . 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 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.
Paper Structure (31 sections, 25 theorems, 159 equations, 5 figures, 3 tables, 4 algorithms)

This paper contains 31 sections, 25 theorems, 159 equations, 5 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Proof of Theorem \ref{['exact_thm']}
  5. Proof of Theorem \ref{['harmonic_thm']}
  6. Proof of Theorem \ref{['hs_thm']}
  7. Multi-type Harris--Sevastyanov transformation
  8. Conditioning on non-extinction at generation t
  9. Population size after the Harris--Sevastyanov transformation
  10. Relationship between harmonic mean at generation 1 and t for the transformed process
  11. Numerical Results
  12. Approximate sampling from the law of $W^{(j)}$
  13. Estimation of Coalescence Probability
  14. Computational cost
  15. Discussion
  16. ...and 16 more sections

Key Result

Theorem 1

Under Assumptions infinite_matrix and R-positive, for all possible founding ancestor types $i_0 \in S$, there exists a real, non-negative random variable $W^{(i_0)}$ with $\mathbb{E}(W^{(i_0)}) = u_{i_0}$ such that as $T \rightarrow \infty$, and, hence, the full population size satisfies

Figures (5)

  • Figure 1: An example realisation of a branching process evolving from generation $0$ to generation $T$ where for the bottom two individuals at time $T$$X_{T, 2} > t$, whereas for the top and bottom individuals $X_{T, 2} < t$.
  • Figure 2: The characteristic function of $W^{(j)}, j \in [2],$ for the system in Example \ref{['poisson']}.
  • Figure 3: Estimated density of $W^{(j)}, j \in [2],$ for the system in Example \ref{['poisson']}.
  • Figure 4: Histograms with their difference for 5000 samples from the approximated density and 5000 renormalised population sizes after 20 generations of a simulated branching process from Example \ref{['poisson']} started from a root ancestor of type 1. Only non-zero values are plotted.
  • Figure 5: Coalescence probability estimation and bounds (see Theorems \ref{['exact_thm']} and \ref{['hs_thm']} respectively).

Theorems & Definitions (50)

  • Theorem 1: Theorem 1 of moy1967
  • Proposition 1: Theorem 7.1 in harris1963
  • Remark
  • Example 1
  • Theorem 2: Theorem 2.1 in Hong2015
  • Theorem 3
  • Proposition 2
  • Remark
  • Remark
  • Theorem 4
  • ...and 40 more