On the survival probability of a branching process in a random environment
Let \(\{Z_n, n \geq 0\}\) be a branching process in an independent and identically distributed random environment; see \textit{K. B. Athreya} and \textit{S. Karlin} [Ann. Math. Stat. 42, 1499-1520, 1843-1858 (1971; Zbl 0228.60032; Zbl 0228.60033)] for details. The author proves that the limit of the survival probability \(P(Z_n > 0)\) as \(n \to \infty\) is 1 if the branching process is supercritical or critical, it is \(E(Z_1)\) if the process is strongly subcritical and it is less than \(\min (1, EZ_1)\) if the process is subcritical but not strongly. The author establishes the result for the case of countable state environment under weaker conditions, thereby extending the work of \textit{F. M. Dekking} [Stochastic Processes Appl. 27, No. 1, 151-157 (1987; Zbl 0634.60072)] pertaining to a finite state environment.
scientific article; zbMATH DE number 847538