Limit Theorems for Supercritical Branching Random Fields

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Abstract
An infinite particle system in Rd is considered where the initial distribution is POISSONian and each initial particle gives rise to a supercritical age‐dependent branching process with the particles moving randomly in space. Our approach differs from the usual: instead of the point measures determined by the locations of the particles at each time, we take the particles at a “final time” and observe the past histories of their ancestry lines. A law of large numbers and a central limit theorem are proved under a space‐time scaling representing high density of particles and small mean particle lifetime. The fluctuation limit is a generalized GAUSS‐MARKOV process with continuous trajectories and satisfies a deterministic evolution equation with generalized random initial condition. A more precise form of the central limit theorem is obtained in the case of particles performing BROWNian motion and having exponentially distributed lifetime.
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