Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

中文翻译：

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在不同环境中分支流程的统一方法

分支过程$(Z_n)_{n \ge 0}$在不同的环境中推广 Galton-Watson 过程，因为它们允许后代分布的时间依赖性。我们的主要结果涉及几乎肯定灭绝的一般标准，鞅的平方可积性$(Z_n/\mathrm E[Z_n])_{n \ge 0}$, 鞅极限的性质W和 Yaglom 类型的结果表明收敛到适当归一化的种群大小的指数极限分布$Z_n$, 以事件为条件$Z_n \gt 0$. 这些定理概括/统一了文献中的不同结果，并导致对过程进行分类。