Asymptotic fluctuations in supercritical Crump–Mode–Jagers processes
Asymptotic fluctuations in supercritical Crump–Mode–Jagers processes
Consider a supercritical Crump--Mode--Jagers process $(\mathcal Z_t^{\varphi})_{t \geq 0}$ counted with random characteristic $\varphi$. Nerman's celebrated law of large numbers [{\it 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, …