On the explosiveness of nonhomogeneous age-dependent branching processes. (English. Russian original) Zbl 0948.60077

Vatutin, V. A.

Theory Probab. Math. Stat. 52, 37-42 (1996); translation from Teor. Jmovirn. Mat. Stat. 52, 37-40 (1995).

The author introduces Bellman-Harris branching processes for which both lifetime and reproduction laws depend on particle generation number. Let \(G_n(t)\) and \(h(n,s)\) be the lifetime distribution and the generating function of offsprings for a particle from the \(n\)th generation, respectively. Let also \(G_{(-n)}(t)\) be the inverse function of \(G_n(t).\) It is shown that positiveness of \(h(n,0)\) and existence of a sequence \(y_n\downarrow 0\) such that \(\sum_{n=1}^{\infty} G_{(-n)}(y_{n-1}/(1-h(n,1-y_n)))<\infty\) imply irregularity of the process.

Reviewer: O.K.Zakusilo (Kyïv)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J85	Applications of branching processes
92D10	Genetics and epigenetics
92D25	Population dynamics (general)

Keywords:

Bellman-Harris branching process; regularity; lifetime; reproduction law