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Asymptotic normality 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). We show that under a certain condition on the intensity of the offspring process, the second-order fluctuations of the age distribution are asymptotically normal; the condition is essential and not just a technicality. This extends to populations counted by a random characteristic.