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Critical multitype branching: a functional limit theorem and ancestral processes

Ellen Baake, Fernando Cordero, Sophia-Marie Mellis, Vitali Wachtel

TL;DR

This work addresses critical multitype branching processes conditioned on non-extinction, analyzing both forward-time trajectories and ancestral histories. It develops a functional limit theorem for the linearly scaled forward process under an $h$-transformed measure, revealing a $4$-dimensional squared Bessel process along the PF right eigenvector and linking the forward limit to Doob's $h$-transform. It then characterizes the ancestral process via a size-biased trunk construction and a retrospective mutation chain, establishing a weak law of large numbers for the ancestral-type distribution with limit $\boldsymbol{\alpha}=(u_i v_i)_{i}$. The paper also provides entrance laws, transition probabilities, and large deviation results for time-averaged ancestral types, yielding a comprehensive forward/backward view of critical multitype branching behavior with explicit probabilistic structures and limit objects. Collectively, these results extend the classical supercritical picture to the critical regime and illuminate the connection between size-biasing, trunk constructions, and ancestral dynamics in multitype populations.

Abstract

We consider the long-term behaviour of critical multitype branching processes conditioned on non-extinction, both with respect to the forward and the ancestral processes. Forward in time, we prove a functional limit theorem in the space of trajectories of the linearly-scaled $h$-transformed process; the change of measure allows us to work on the same probability space for all times. Backward in time, we trace the lines of descent of individuals sampled from a (stationary) population and analyse the ancestral type distribution, that is, the type distribution of the ancestors in the distant past, as well as the type process along the ancestral line.

Critical multitype branching: a functional limit theorem and ancestral processes

TL;DR

This work addresses critical multitype branching processes conditioned on non-extinction, analyzing both forward-time trajectories and ancestral histories. It develops a functional limit theorem for the linearly scaled forward process under an -transformed measure, revealing a -dimensional squared Bessel process along the PF right eigenvector and linking the forward limit to Doob's -transform. It then characterizes the ancestral process via a size-biased trunk construction and a retrospective mutation chain, establishing a weak law of large numbers for the ancestral-type distribution with limit . The paper also provides entrance laws, transition probabilities, and large deviation results for time-averaged ancestral types, yielding a comprehensive forward/backward view of critical multitype branching behavior with explicit probabilistic structures and limit objects. Collectively, these results extend the classical supercritical picture to the critical regime and illuminate the connection between size-biasing, trunk constructions, and ancestral dynamics in multitype populations.

Abstract

We consider the long-term behaviour of critical multitype branching processes conditioned on non-extinction, both with respect to the forward and the ancestral processes. Forward in time, we prove a functional limit theorem in the space of trajectories of the linearly-scaled -transformed process; the change of measure allows us to work on the same probability space for all times. Backward in time, we trace the lines of descent of individuals sampled from a (stationary) population and analyse the ancestral type distribution, that is, the type distribution of the ancestors in the distant past, as well as the type process along the ancestral line.
Paper Structure (12 sections, 15 theorems, 178 equations, 3 figures)

This paper contains 12 sections, 15 theorems, 178 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The process and the family tree
  4. The measure $\widehat{\mathbb{P}}$
  5. A functional limit theorem
  6. Fdd convergence
  7. Tightness
  8. Population average of ancestral types
  9. Construction of the size-biased family tree
  10. Time average of ancestral types
  11. Typical ancestral type evolution
  12. Acknowledgements

Key Result

Theorem 3.1

Let $(B(t))_{t \geq 0}$ be a $4$-dimensional squared Bessel process started at $B(0)=0$. Assume that the third moments of the offspring distribution are finite and set, for $T>0$, Then, under $\widehat{\mathbb{P}}_{\boldsymbol{z}}$ and for any $\boldsymbol{z} \in \mathbb{N}_{0}^{d}\setminus \{ \boldsymbol{0}\}$, $\boldsymbol{Y}^{(n)}$ converges in distribution, as $n \to \infty$, to $\boldsymbol{

Figures (3)

  • Figure 1: A family tree for $d=3$. Types are indicated by different colours, indexed in the order (blue, magenta, green), counted from top to bottom. We have distinguished the 3rd-generation individual $x=(\textcolor{navyblue}{blue}, 1; \textcolor{darklavender!70}{ magenta}, 1; \textcolor{darkgreen!60}{ green}, 1)$ (that is, the circled edge intersecting the vertical dashed line at $3$). Its unique ancestor in generation 1 is $x(1)=(\textcolor{navyblue}{blue}, 1)$; $X(x,5)$ (that is, the set of all edges that emanate from $x$ and hit the vertical dashed line at $5$) is its set of descendants in generation 5, and $\boldsymbol{Z}(5)$ counts the type frequencies in the population $X(5)$ (that is, all edges that intersect the vertical dashed line at $5$).
  • Figure 2: A familiy tree for $d=3$, where $A^{(1)}_{\textcolor{navyblue}{blue}}(5)$ is highlighted in bold.
  • Figure 3: A size-biased Galton-Watson tree for $d=3$. The type process on the trunk corresponds to the retrospective mutation chain, while the collection of bushes (though still descendants of the trunk, as indicated by the dotted lines) may be understood as branching processes with immigration depending on the current state of the trunk.

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4: Entrance law
  • proof
  • ...and 29 more