Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes

Consider a supercritical Crump–Mode–Jagers process $(\mathcal Z_t^{\varphi})_{t \geq 0}$countedwitharandomcharacteristic$\varphi$. Nerman’s celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365–395, 1981] states that, under some mild assumptions, $e^{-\alpha t} \mathcal Z_t^\varphi$convergesalmostsurelyas$t \to \infty$to$aW$\mathrm{.}Here,$\alpha>0$istheMalthusianparameter,$a$isaconstantand$W$ 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 $({\mathcal{Z}}_t^{\phi }{)}_{t\ge 0}$. 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 $\alpha \mathrm{/}2\le \mathrm{R}\mathrm{e}(\lambda )<\alpha$, such that $({\mathcal{Z}}_t^{\phi }-a{e}^{\alpha t}W-H(t))\mathrm{/}\sqrt{t^k{e}^{\alpha t}}$ 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.