Multitype branching processes in random environments

Abstract

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.