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On the almost sure convergence of controlled branching processes. (English) Zbl 0608.60079
Let $$X(n+1)=Z_ 1(n)+...+Z_{\phi (x(n))}(n)$$ be the population size at time $$n+1$$ for a controlled branching process. Here $$Z_ j(n)$$ is the number of off-spring of individual j in the nth generation and $$\phi$$ is a control function which satisfies conditions similar to those of A. M. Zubkov [Teor. Veroyatn. Primen. 19, 319-339 (1974; Zbl 0321.60063)], typically $$\phi$$ ($$\ell)\simeq A\ell$$, $$\ell \to \infty$$. With $$\mu =EZ$$ the offspring mean, it is shown that $$X(n)/(\mu A)^ n$$ converges a.s. to a limit X with $$X\neq 0$$ if and only if EZ log Z$$<\infty.$$
The method of proof is that of the reviewer [Branching processes, Conf. Quebec 1976, Adv. Probab. relat. Top., Vol. 5, 1-26 (1978; Zbl 0402.60083)].
Reviewer: S.Asmussen

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