Beyond Galton-Watson Processes: Forests, Duals, and Ranks

Abstract

A random forest is a random graph (V,E) with a set of vertices V = N20 and a set of edges E = {ev, v 2 V } satisfying the following property: if v = (x, t + 1), then ev = (v, v0), where v0 = (x0, t) and x0 = 't(x) is an increasing stochastic process in x. For a given forest, there is a unique way to draw a dual forest. These forests can be used as a graphical representation of discrete time reproduction processes forward and backward in time. They also serve to introduce a new concept, ranked Galton-Watson processes, where individual reproduction depends on the position in the population. A main result is that the dual process to a Galton-Watson process in varying environments with immigration is a Galton-Watson process in varying environments if and only if the reproduction and immigration laws of the first process are linear fractional.