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We consider an age-dependent branching particle system in ℝd,  where the particles are subject to α-stable migration  (0 < α ≤ 2), critical binary branching, and  general (non-arithmetic) lifetimes distribution. The population starts off from a Poisson  random field in ℝd with Lebesgue intensity. We prove  functional central limit theorems and strong laws of large numbers under two rescalings:  high particle density, and a space-time rescaling that preserves the migration  distribution. Properties of the limit processes such as Markov property, almost sure  continuity of paths and generalized Langevin equation, are also investigated

Fluctuation limit theorems for age-dependent critical binary branching systems