MULTITYPE BRANCHING PROCESSES IN RANDOM ENVIRONMENTS,

Consider the k-type Galton-Watson process. The work removes the restrictive assumption that particles of the same type always divide in accordance with the same probability generating function p.g.f.. Instead, it is assumed that at each unit of time, Nature chooses a k-vector of p.g.f.s from a class of k-vectors of 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 p.g.f. assigned to its classification. This process is called a multitype branching process in a random environment MBPRE. The work gives some necessary and some sufficient conditions for almost certain extinction of the MBPRE when there are at least two particle classifications. To obtain these results, Jensens inequality, the dual process suggested by Smith and Wilkinson, and some results on products of random matrices are used. Since the theorems require the user to evaluate limits of sequences of products of random arrays, some corollaries are included which involve simpler, albeit less general, conditions. Author