Asymptotic fluctuations of supercritical general branching processes

研究了超临界单型和多型一般（Crump-Mode-Jagers）分支过程。目标是精确地理解它们在大数定律之外的时间渐近行为。困难来自于这样一个事实，即在某些情况下，大数定律和中心极限定理的尺度之间会出现周期性波动。这些周期性与复鞅和复平滑方程的收敛性密切相关。方法来自分支过程、随机过程的收敛、鞅理论、更新和马尔可夫更新理论、拉普拉斯变换和复平滑方程理论。

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.