Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes

TL;DR

This paper establishes a central limit theorem for supercritical Crump-Mode-Jagers processes, describing the asymptotic fluctuations of the process around its almost sure limit, extending previous results in branching process theory.

Contribution

It proves a general CLT for these processes, unifying and extending existing specific results, under mild and second moment assumptions.

Findings

Convergence of normalized process to a normal distribution.

Identification of a finite random linear combination H(t) capturing main fluctuations.

Extension of CLT results to a broad class of branching processes.

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, $e^{-\alpha t} \mathcal Z_t^\varphi$ converges almost surely as $t \to \infty$ to $aW$. Here, $\alpha>0$ 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 $t^j e^{\lambda t}$ with $\alpha/2 \leq \mathrm{Re}(\lambda)<\alpha$, such that $(\mathcal Z_t^\varphi - a e^{\alpha t}W -H(t))/\sqrt{t^k e^{\alpha t}}$ 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.