Asymptotic fluctuations in supercritical Crump–Mode–Jagers processes

Abstract

Consider a supercritical Crump–Mode–Jagers process ${({\mathcal{Z}}_{\mathit{t}}^{\mathit{\phi }})}_{\mathit{t}\ge 0}$ counted with a random characteristic φ. Nerman’s celebrated law of large numbers (Z. Wahrsch. Verw. Gebiete 57 (1981) 365–395) states that, under some mild assumptions, ${\mathit{e}}^{-\mathit{\alpha }\mathit{t}}{\mathcal{Z}}_{\mathit{t}}^{\mathit{\phi }}$ converges almost surely as $\mathit{t}\to \infty$ to $\mathit{a}\mathit{W}$. Here, $\mathit{\alpha }>0$ 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 ${({\mathcal{Z}}_{\mathit{t}}^{\mathit{\phi }})}_{\mathit{t}\ge 0}$. More precisely, we show that there exist a constant $\mathit{k}\in {\mathbb{N}}_0$ and a function $\mathit{H}(\mathit{t})$, a finite random linear combination of functions of the form ${\mathit{t}}^{\mathit{j}}{\mathit{e}}^{\mathit{\lambda }\mathit{t}}$ with $\mathit{\alpha }/2\le Re(\mathit{\lambda })<\mathit{\alpha }$, such that $({\mathcal{Z}}_{\mathit{t}}^{\mathit{\phi }}-\mathit{a}{\mathit{e}}^{\mathit{\alpha }\mathit{t}}\mathit{W}-\mathit{H}(\mathit{t}))/\sqrt{{\mathit{t}}^{\mathit{k}}{\mathit{e}}^{\mathit{\alpha }\mathit{t}}}$ 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.