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On the strong law of large numbers and Llog L condition for supercritical general branching processes

Vincent Bansaye, Tresnia Berah, Bertrand Cloez

TL;DR

The paper develops a unified Kesten–Stigum-type law of large numbers for supercritical general branching processes with trait structure in infinite-type spaces. By introducing a first moment semigroup that admits a contraction in a weighted total variation sense under a Lyapunov framework, and by enforcing an $L\log L$ moment condition, it proves almost sure and $L^1$ convergence of renormalized empirical measures to a nondegenerate martingale limit $W$, with $Z_n(f)/\lambda^n \to \gamma(f)W$ for suitable observables. The results extend classical finite-type LLN to infinite dimensions, relax uniform convergence of the renormalized first moment, and provide a robust martingale decomposition via truncation to handle general state spaces. The framework is then applied to continuous-time settings and diverse models (branching elliptic diffusions, House of Cards, growth-fragmentation), illustrating broad applicability and unifying prior scattered LLN results under minimal moment assumptions. This advances the understanding of long-term collective behavior in structured populations and offers practical tools for analyzing complex branching systems.

Abstract

We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the normalized empirical measure and non-degeneracy of the limiting martingale. Convergence is obtained under an Llog L condition which provides a Kesten-Stigum result in infinite dimension and relaxes the uniform convergence assumption of the renormalized first moment semigroup required in the work of Asmussen and Hering in 1976. The techniques of proofs combine families of martingales and contraction of semigroups and the truncation procedure of Asmussen and Hering. We also obtain L^1 convergence of the renormalized empirical measure and contribute to unifying different results in the literature. These results greatly extend the class of examples where a law of large numbers applies, as we illustrate it with absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.

On the strong law of large numbers and Llog L condition for supercritical general branching processes

TL;DR

The paper develops a unified Kesten–Stigum-type law of large numbers for supercritical general branching processes with trait structure in infinite-type spaces. By introducing a first moment semigroup that admits a contraction in a weighted total variation sense under a Lyapunov framework, and by enforcing an moment condition, it proves almost sure and convergence of renormalized empirical measures to a nondegenerate martingale limit , with for suitable observables. The results extend classical finite-type LLN to infinite dimensions, relax uniform convergence of the renormalized first moment, and provide a robust martingale decomposition via truncation to handle general state spaces. The framework is then applied to continuous-time settings and diverse models (branching elliptic diffusions, House of Cards, growth-fragmentation), illustrating broad applicability and unifying prior scattered LLN results under minimal moment assumptions. This advances the understanding of long-term collective behavior in structured populations and offers practical tools for analyzing complex branching systems.

Abstract

We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the normalized empirical measure and non-degeneracy of the limiting martingale. Convergence is obtained under an Llog L condition which provides a Kesten-Stigum result in infinite dimension and relaxes the uniform convergence assumption of the renormalized first moment semigroup required in the work of Asmussen and Hering in 1976. The techniques of proofs combine families of martingales and contraction of semigroups and the truncation procedure of Asmussen and Hering. We also obtain L^1 convergence of the renormalized empirical measure and contribute to unifying different results in the literature. These results greatly extend the class of examples where a law of large numbers applies, as we illustrate it with absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.

Paper Structure

This paper contains 18 sections, 16 theorems, 214 equations.

Table of Contents

  1. Introduction and main results
  2. General study in discrete time
  3. Construction and definitions
  4. Contraction and $L\log L$ moment
  5. Families of martingales
  6. Proof of Theorem \ref{['th:main-discret']}
  7. Convergence and applications in continuous time
  8. Main result
  9. Construction and preliminaries for applications
  10. The trait dynamics.
  11. The branching mechanism.
  12. Non-explosion and generator.
  13. Application to branching elliptic diffusion
  14. Description of the branching Markov process
  15. Asymptotic behaviour.
  16. ...and 3 more sections

Key Result

Theorem 1.2

Under Assumption ass:cvSt, we further suppose that there exists a measurable function $V: \mathcal{X} \to (0,\infty)$ such that $\sup_{\mathcal{X}}h/V<\infty$, $V\leq V^\star$ and for any $k\in \mathbb{N}^*$ Then, for any $x\in \mathcal{X}$, Besides, for any $f$ such that $\sup_{\mathcal{X}}f/V<\infty$,

Theorems & Definitions (32)

  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 22 more