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Sharp $L \log L$ condition for supercritical Galton-Watson processes with countable types

Mathilde André, Jean-Jil Duchamps

TL;DR

The paper develops a sharp $L log L$ criterion for supercritical Galton–Watson processes with a countable set of types under positive recurrence of the mean matrix, extending Kesten–Stigum-type results to infinite-type settings. Using a spinal change of measure and a spine-biased construction, it characterizes when the additive martingale is uniformly integrable and links this to extinction properties and the nondegeneracy of the limit $W$. It then proves convergence of the type distribution: in the positive recurrent case, $\rho^{-n}Z(n)$ converges in probability to a multiple of the left eigenvector, and under additional moment and entropy conditions, convergence holds almost surely; in the null recurrent case, only weaker, ergodic-in-mean results are obtained. The work also provides explicit examples and counter-examples illustrating the necessity and limits of the proposed conditions and discusses implications for biological population models and network epidemics. These results significantly extend Kesten–Stigum theory to broad, countable-type branching structures with potential infinite population sizes per generation.

Abstract

We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp $L\log L$ condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

Sharp $L \log L$ condition for supercritical Galton-Watson processes with countable types

TL;DR

The paper develops a sharp criterion for supercritical Galton–Watson processes with a countable set of types under positive recurrence of the mean matrix, extending Kesten–Stigum-type results to infinite-type settings. Using a spinal change of measure and a spine-biased construction, it characterizes when the additive martingale is uniformly integrable and links this to extinction properties and the nondegeneracy of the limit . It then proves convergence of the type distribution: in the positive recurrent case, converges in probability to a multiple of the left eigenvector, and under additional moment and entropy conditions, convergence holds almost surely; in the null recurrent case, only weaker, ergodic-in-mean results are obtained. The work also provides explicit examples and counter-examples illustrating the necessity and limits of the proposed conditions and discusses implications for biological population models and network epidemics. These results significantly extend Kesten–Stigum theory to broad, countable-type branching structures with potential infinite population sizes per generation.

Abstract

We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

Paper Structure

This paper contains 11 sections, 12 theorems, 73 equations.

Table of Contents

  1. Introduction and main results
  2. Examples and counter-examples
  3. Condition \ref{['eq:simple_condition']} may fail while \ref{['eq:nsc']} holds, even in the positive recurrent case.
  4. Extinction is not characterized by $W=0$, even in the positive recurrent case.
  5. Motivation and applications to population biology.
  6. Proof of Theorem \ref{['thm:ks_condition']}
  7. Proof of Theorem \ref{['thm:type_convergence']}
  8. Ergodicity in mean.
  9. \ref{['ass:variance_condition']}: Offspring distributions with finite variance.
  10. \ref{['ass:entropy_condition']}: Entropy condition.
  11. Acknowledgements.

Key Result

Proposition 1

Assume $M$ is irreducible, aperiodic and recurrent, with a finite Perron value $\varrho$. Then $\varrho$ is the largest eigenvalue of $M$ and is simple. Moreover, there exists associated left and right eigenvectors $\tilde{h}\in\mathop{\mathrm{\mathbb{R}}}\nolimits^{\mathcal{X} *}$ and $h\in\mathop{ Otherwise (i.e. when $\tilde{h}h=\infty$), $M$ is said to be null recurrent and we have $\lim_{n\t

Theorems & Definitions (22)

  • Proposition 1: Extension of Perron--Frobenius theory to infinite matrices
  • Theorem 2: Uniform integrability of the additive martingale
  • Remark 3: Extinction of the branching process
  • Remark 4
  • Theorem 5: Convergence of the type distribution
  • Remark 6
  • Remark 7
  • Lemma 8
  • proof
  • Corollary 9
  • ...and 12 more