Asymptotic fluctuations of supercritical general branching processes

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We study supercritical single- and multi-type general (Crump-Mode-Jagers) branching processes. The goal is to precisely understand their asymptotic bevahiour in time beyond the law of large numbers. The difficulty comes from the fact that in certain cases, periodic fluctuations between the scales of the law of large numbers and the central limit theorem arise. These periodicities are intimately connected with convergence of complex martingales and complex smoothing equations. Methods come from the fields of branching processes, convergence of stochastic processes, martingale theory, renewal and Markov renewal theory, Laplace transformation, and the theory of complex smoothing equations.