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Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes

Alexander Iksanov, Konrad Kolesko, Matthias Meiners

Abstract

Consider a supercritical Crump--Mode--Jagers process $(\mathcal Z_t^{\varphi})_{t \geq 0}$ counted with a random characteristic $\varphi$. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, $e^{-αt} \mathcal Z_t^\varphi$ converges almost surely as $t \to \infty$ to $aW$. Here, $α>0$ is the Malthusian parameter, $a$ is a constant and $W$ is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional (second moment) assumptions, we prove a central limit theorem for $(\mathcal Z_t^{\varphi})_{t \geq 0}$. More precisely, we show that there exist a constant $k \in \mathbb N_0$ and a function $H(t)$, a finite random linear combination of functions of the form $t^j e^{λt}$ with $α/2 \leq \mathrm{Re}(λ)<α$, such that $(\mathcal Z_t^\varphi - a e^{αt}W -H(t))/\sqrt{t^k e^{αt}}$ converges in distribution to a normal random variable with random variance. This result unifies and extends various central limit theorem-type results for specific branching processes.

Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes

Abstract

Consider a supercritical Crump--Mode--Jagers process counted with a random characteristic . Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, converges almost surely as to . Here, is the Malthusian parameter, is a constant and is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional (second moment) assumptions, we prove a central limit theorem for . More precisely, we show that there exist a constant and a function , a finite random linear combination of functions of the form with , such that converges in distribution to a normal random variable with random variance. This result unifies and extends various central limit theorem-type results for specific branching processes.

Paper Structure

This paper contains 30 sections, 25 theorems, 362 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup, preliminaries and main results
  3. The general branching process counted with a random characteristic
  4. Assumptions
  5. Main results
  6. Applications
  7. The Galton-Watson process
  8. Nerman's martingales
  9. Epidemic models
  10. Supercritical binary homogeneous Crump-Mode-Jagers processes
  11. The conservative fragmentation model
  12. Preliminaries for the proofs of the main results
  13. Change of measure and the connection to renewal theory
  14. The expectation of the general branching process
  15. Matrix notation
  16. ...and 15 more sections

Key Result

Proposition 2.2

Suppose that (Aass:Malthusian parameter) holds and that $\varphi$ is a random characteristic satisfying (Aass:mean growth) and (Aass:variance growth). Then, for every $t \in \mathds{R}$, converges unconditionally in $L^1$ and almost surely over every admissible ordering of $\mathcal{I}$.

Figures (3)

  • Figure 1: The subtree $\mathcal{I}_{16}$ for the particular admissible ordering of $\mathcal{I}$ given above.
  • Figure 2: Diagram representing the dependence of the results.
  • Figure 3: Solutions to $\mathcal{L}\mu(\lambda)=1$ in the cases $a=18$, $b=1$, $R_0=10$ (left figure) and $a=18$, $b=1$, $R_0=12$. In the left figure, the root $\lambda \not = \alpha$ with largest real part has $\mathrm{Re}(\lambda) < \frac{\alpha}{2}$, in the right figure $\mathrm{Re}(\lambda) > \frac{\alpha}{2}$.

Theorems & Definitions (69)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 59 more