Asymptotics of fluctuations in Crump-Mode-Jagers processes: the lattice case

Consider a supercritical Crump--Mode--Jagers process such that all births are at integer times (the lattice case). Let $\widehat\mu(z)$ be the generating function of the intensity of the offspring process, and consider the complex roots of $\widehat\mu(z)=1$. The smallest (in absolute value) such root is $e^{-\alpha}$, where $\alpha>0$ is the Malthusian parameter; let $\gamma_*$ be the second smallest absolute value of a root. We show, assuming some technical conditions, that there are three cases: (i) if $\gamma_*>e^{-\alpha/2}$, then the second-order fluctuations of the age distribution are asymptotically normal; (ii) if $\gamma_*=e^{-\alpha/2}$, then the fluctuations are still asymptotically normal, but with a larger order of the variance; (iii) if $\gamma_*<e^{-\alpha/2}$, then the fluctuations are even larger, but will oscillate and (except in degenerate cases) not converge in distribution. This trichotomy is similar to what has been seen in related situations, e.g. for some other branching processes, and for P\'olya urns. The results lead to a symbolic calculus describing the limits. The results extends to populations counted by a random characteristic.