BRANCHING PROCESSES IN STOCHASTIC ENVIRONMENTS,

This paper is concerned with two simple models for branching processes in stochastic environments. The models are identical to the model for the classical Galton-Watson branching process in all respects but one. In the family-tree language commonly used to describe branching processes, that difference is that the probability distribution for the number of offspring of an object changes stochastically from one generation to the next, and is the same for all members of the same generation. That is, the probability distribution of the number of offspring is a function of the environment. The two models considered herein have a random environment and a Markov environment. The object of study is the probability distribution of the number of objects Zn in the nth generation. Of particular interest is the determination of conditions under which the family has probability one of dying out.