Abstract
A well established fact about supercritical branching population processes is that they either become extinct or explode at an exponential rate. Still it is possible to imagine some kind of asymptotic stability even in this case, namely that of the composition, in several senses, of a population according to some appropriate limit composition laws, even though the population size continues to grow. This idea has been exploited by many authors. A very useful way of thinking about such composition laws is to view them as related to some randomly sampled individual chosen in some convenient manner from a very old branching population (see below). Probabilistic statements concerning different aspects of the pedigree of this sampled individual can then be done using the probability measure P the stable pedigree measure, which in a sense summarizes the above mentioned limit laws, and which is induced by the sampling mechanism and the life law. In the present chapter we make some definitions and introduce some notation to be used in the coming chapters as a straightforward tool in the study of retrospective aspects of labelled branching processes. The whole chapter is more or less based on [Jagers & Nerman 1984a] and [Nerman & Jagers, 1984].
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