General branching processes in discrete time as random trees.

Journal article, 2008

The simple Galton-Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory. We show how a similar reinterpretation can shed new light on the more interesting forms of branching processes that allow repeated bearings and, thus, overlapping generations. In particular, we use the stable pedigree law to give a transparent description of a size-biased version of general branching processes in discrete time. This allows us to analyse the xlog x condition for exponential growth of supercritical general processes, and also the relation between simple Galton-Watson and more general branching processes.