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Central limit theorem for supercritical Crump-Mode-Jagers processes counted with non-individual random characteristics

Gabriel Berzunza Ojeda, Harlan Connor

TL;DR

The paper proves a central limit theorem for general CMJ-branching processes counted with random characteristics that may depend on individuals’ lifespans and their descendants up to a fixed generation. By developing an explicit martingale-based CLT for centred and non-centred descendant characteristics, and leveraging renewal theory with a Malthusian parameter \(\alpha\) and Nerman’s martingale \(W\), the authors derive stable convergence to a Gaussian limit with variance given by a renewal-integral expression involving the centred component \(\chi^{(\varphi,h)}\). The results extend prior work by relaxing independence across individuals and provide a thorough framework for fringe-tree asymptotics, including explicit conditions under which the fringe-tree counts are asymptotically normal (via Theorem \ref{Theo2} and Corollary \ref{corollary5}). This advances both the theory of general CMJ processes and applications to random trees, answering open questions on fringe-tree fluctuations. The methodology combines advanced branching-process martingales, renewal theory, and careful handling of dependencies across generations to establish stable CLTs with explicit variance formulas.

Abstract

Consider a supercritical Crump-Mode-Jagers process $(\mathcal{Z}_{t}^{\varphi})_{t \geq 0}$ counted with a random characteristic $\varphi$ that depends on an individual's life and their descendant process up to a fixed generation. Under second moment assumptions, we establish a central limit theorem for $\mathcal{Z}_{t}^{\varphi}$ as $t \rightarrow \infty$. Our result extends the recent work of Iksanov, Kolesko, and Meiners (2024) by relaxing their assumption of independent characteristics across individuals. We further demonstrate the applicability of our results to the study of fringe trees in several important random tree families, thereby providing insights into questions raised by Holmgren and Janson (2017).

Central limit theorem for supercritical Crump-Mode-Jagers processes counted with non-individual random characteristics

TL;DR

The paper proves a central limit theorem for general CMJ-branching processes counted with random characteristics that may depend on individuals’ lifespans and their descendants up to a fixed generation. By developing an explicit martingale-based CLT for centred and non-centred descendant characteristics, and leveraging renewal theory with a Malthusian parameter and Nerman’s martingale , the authors derive stable convergence to a Gaussian limit with variance given by a renewal-integral expression involving the centred component \(\chi^{(\varphi,h)}\). The results extend prior work by relaxing independence across individuals and provide a thorough framework for fringe-tree asymptotics, including explicit conditions under which the fringe-tree counts are asymptotically normal (via Theorem \ref{Theo2} and Corollary \ref{corollary5}). This advances both the theory of general CMJ processes and applications to random trees, answering open questions on fringe-tree fluctuations. The methodology combines advanced branching-process martingales, renewal theory, and careful handling of dependencies across generations to establish stable CLTs with explicit variance formulas.

Abstract

Consider a supercritical Crump-Mode-Jagers process counted with a random characteristic that depends on an individual's life and their descendant process up to a fixed generation. Under second moment assumptions, we establish a central limit theorem for as . Our result extends the recent work of Iksanov, Kolesko, and Meiners (2024) by relaxing their assumption of independent characteristics across individuals. We further demonstrate the applicability of our results to the study of fringe trees in several important random tree families, thereby providing insights into questions raised by Holmgren and Janson (2017).

Paper Structure

This paper contains 23 sections, 25 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Notation.
  3. The CMJ-branching process and characteristics
  4. The main results
  5. Assumptions on the birth point process
  6. Assumptions on the random characteristics
  7. Main results
  8. Application to fringe trees
  9. Auxiliary results
  10. The expectation of the general branching process
  11. Biggins' martingale
  12. Conditional expectations
  13. Proof of Proposition \ref{['Proposition3']}
  14. Properties of the characteristic $\chi^{(\varphi, h)}$
  15. A central limit theorem for centred characteristics
  16. ...and 8 more sections

Key Result

Proposition 3.3

Suppose that A1-A7 hold and that $\varphi$ is an $h$-dependent descendant characteristic, for some $h \in \mathbb{N}_{0}$, satisfying C1-C3. Then, for every $t \in \mathbb{R}$, converges unconditionally in $L^{1}(\Omega, \mathcal{F}, \mathbb{P})$ and almost surely over every admissible ordering of $\mathcal{I}$.

Theorems & Definitions (65)

  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Lemma 3.7
  • Theorem 3.8
  • Corollary 3.9
  • Remark 3.10
  • ...and 55 more