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Asymptotic behaviour of critical controlled branching processes with random control functions. (English) Zbl 1079.60073
Let \(Z_{n+1} = \sum^{\varphi_n (Z_n)} X_{n, j}\) be a controlled branching process with i.i.d. random control functions \(\varphi_n\). In the critical case, \(E(Z_{n+1}/Z_n\mid Z_n = k) \rightarrow_k 1\), the authors establish results on extinction, nonextinction and of the limiting distribution under suitable normalization. The main tool are methods for growth processes \(Z_{n+1} = Z_n + g (Z_n) \xi_n\) as in [G. Keller, G. Kersting and the reviewer, Ann. Prob. 15, 305–343 (1987; Zbl 0616.60079)].
Reviewer: Uwe Rösler (Kiel)

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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