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A Kesten Stigum theorem for Galton-Watson processes with infinitely many types in a random environment

Maxime Ligonnière

TL;DR

The paper develops a Kesten–Stigum-type framework for a multitype Galton–Watson process with infinitely many types in a random ergodic environment, establishing a fundamental martingale built from space-time harmonic eigenfunctions and proving its nondegeneracy under an $L\log L$-type condition. It provides a detailed asymptotic description of the surviving population through the quenched mean and shows that, on the event of nontrivial limit $W>0$, population sizes and type distributions are governed by the quenched mean at large times. The results extend to an age-structured Leslie-type GWRE (Leslie-GWRE), with tractable criteria ensuring the core hypotheses and explicit criteria for extinction vs. explosion. Taken together, the work advances understanding of survival, growth, and type-frequency behavior in MGWREs with infinitely many types and random environments, and offers concrete tools for age-structured applications.

Abstract

In this paper, we study a Galton-Watson process $(Z_n)$ with infinitely many types in a random ergodic environment $\barξ=(ξ_n)_{n\geq 0}$. We focus on the supercritical regime of the process, where the quenched average of the size of the population grows exponentially fast to infinity. We work under Doeblin-type assumptions coming from a previous paper, which ensure that the quenched mean semi group of $(Z_n)$ satisfies some ergodicity property and admits a $\barξ$-measurable family of space-time harmonic functions. We use these properties to derive an associated nonnegative martingale $(W_n)$. Under a $L\log(L)^{1+\varepsilon}$-integrabilty assumption on the offspring distribution, we prove that the almost sure limit $W$ of the martingale $(W_n)$ is not degenerate. Assuming some uniform $L^2$-integrability of the offspring distribution, we prove that conditionally on $\{W>0\}$, at a large time $n$, both the size of the population and the distribution of types correspond to those of the quenched mean of the population $\mathbb{E}[Z_n|\barξ, Z_0]$. We finally introduce an example of a process modelling a population with a discrete age structure. In this context, we provide more tractable criterions which guarantee our various assumptions are met.

A Kesten Stigum theorem for Galton-Watson processes with infinitely many types in a random environment

TL;DR

The paper develops a Kesten–Stigum-type framework for a multitype Galton–Watson process with infinitely many types in a random ergodic environment, establishing a fundamental martingale built from space-time harmonic eigenfunctions and proving its nondegeneracy under an -type condition. It provides a detailed asymptotic description of the surviving population through the quenched mean and shows that, on the event of nontrivial limit , population sizes and type distributions are governed by the quenched mean at large times. The results extend to an age-structured Leslie-type GWRE (Leslie-GWRE), with tractable criteria ensuring the core hypotheses and explicit criteria for extinction vs. explosion. Taken together, the work advances understanding of survival, growth, and type-frequency behavior in MGWREs with infinitely many types and random environments, and offers concrete tools for age-structured applications.

Abstract

In this paper, we study a Galton-Watson process with infinitely many types in a random ergodic environment . We focus on the supercritical regime of the process, where the quenched average of the size of the population grows exponentially fast to infinity. We work under Doeblin-type assumptions coming from a previous paper, which ensure that the quenched mean semi group of satisfies some ergodicity property and admits a -measurable family of space-time harmonic functions. We use these properties to derive an associated nonnegative martingale . Under a -integrabilty assumption on the offspring distribution, we prove that the almost sure limit of the martingale is not degenerate. Assuming some uniform -integrability of the offspring distribution, we prove that conditionally on , at a large time , both the size of the population and the distribution of types correspond to those of the quenched mean of the population . We finally introduce an example of a process modelling a population with a discrete age structure. In this context, we provide more tractable criterions which guarantee our various assumptions are met.

Paper Structure

This paper contains 25 sections, 21 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminary assumptions
  3. Results
  4. The fundamental martingale
  5. Non degeneracy of the martingale
  6. Asymptotic type distribution
  7. Extinction and explosion
  8. Structure of the paper
  9. Preliminaries : behavior of the quenched mean of the process
  10. Proofs
  11. The fundamental martingale
  12. Condition for nondegeneracy of the fundamental martingale
  13. Construction of the Galton-Watson tree
  14. Construction of the size biased tree
  15. Nondegeneracy of the martingale
  16. ...and 10 more sections

Key Result

Theorem 1

Consider a MGWRE satisfying assumptions ass: boundedness_KS to ass : moments_cd_KS. Then, conditionally on the environmental process $(\xi_n)_{n\geq 0}$, the process $(W_n)$ is a $({\mathcal{F}}_n)_{n\geq 0}$-martingale.

Theorems & Definitions (43)

  • Definition 1.1
  • Theorem 1: The fundamental martingale
  • Theorem 2: Non-degeneracy of the fundamental martingale
  • Remark 1.1
  • Theorem 3
  • Corollary 1.2
  • Proposition 1.3
  • Theorem A: ligonniere_ergodic_2023
  • Corollary 2.1
  • Lemma 3.1
  • ...and 33 more