We consider an age-dependent branching particle system in ?d, 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 ?d 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.
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