In this paper, growth of branching processes in random environment is considered. In particular it is shown that this process either "explodes" at an exponential rate or else becomes extinct w.p.1. A classification theorem outlining the cases of "explosion or extinction" is given. To prove these theorems, the associated branching process (the process conditioned on each particle having infinite descent) and the reduced branching process (the particles of the process having infinite descent) are introduced. The method of proof used, in general, is direct probabilistic computation, in contrast with the classical functional iteration method.
"Limit Theorems for Branching Processes in a Random Environment." Ann. Probab. 5 (1) 100 - 116, February, 1977. https://doi.org/10.1214/aop/1176995894