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A two-type Bellman-Harris process initiated by a large number of particles. (English) Zbl 1321.60180

Summary: We investigate a two-type critical Bellman-Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order \(o(t^{-2})\); the tail of the life-length distribution of the second type particles is regularly varying at infinity with index \(-\beta\), \(\beta \in(0,1]\); at time \(t=0\) the process starts with a large number \(N\) of the second type particles and no particles of the first type. It is shown that the time axis \(0 \leq t < \infty\) splits into several regions whose ranges depend on \(\beta\) and the ratio \(N/t\) within each of which the process at time \(t\) exhibits asymptotics (as \(N\), \(t \to \infty\)) which are different from those in the other regions.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K05 Renewal theory
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References:

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