Multitype Branching Processes in Random Environments

Consider a population of particles and k fixed particle classifications labeled type 1, type 2, ... , type k. Suppose that to each classification there corresponds exactly one k-variate p.g.f. and that each particle of the population is classified by one and only one type. Further, assume that after each unit of time, each particle, independently of the other particles, splits or disintegrates into particles of several types, in accordance with the k-variate p.g.f. which corresponds to the parent particle's type. Given these hypotheses, the probability of the population ever becoming extinct is known. This process, in fact, is the well-known multi type Galton-Watson process. In this work, we remove the restrictive assumption that particles of the same type always divide in accordance with the same k-variate p.g.f. Instead, we assume that at each unit of time, Nature chooses a k-vector of k-variate p.g.f.s from a class of k-vectors of k-variate p.g.f.s, independently of the population, past and present, and the previously selected k-vectors, which is then assigned to the present population. Each particle of the present population then splits or disintegrates, independently of the others, in accordance with the k-variate p.g.f. assigned to its