Geodesic
On the Structure of Spatial Branching Processes
Serdica Mathematical Journal, Tome 23 (1997) no. 3-4, pp. 269-312.

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The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk) k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure.
Keywords: Branching Particle Systems, Two-Sided Infinite Markov Sequences of a Random Populations, Genealogy, Poisson Distribution 
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Matthes, Klaus; Nawrotzki, Kurt; Siegmund-Schultze, Rainer. On the Structure of Spatial Branching Processes. Serdica Mathematical Journal, Tome 23 (1997) no. 3-4, pp. 269-312. http://geodesic.mathdoc.fr/item/SMJ2_1997_23_3-4_a7/