Sibling Dependences in Branching Populations

Broberg, P. (1987). Sibling Dependences in Branching Populations. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-87-059

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Abstract

The branching process got its start with a demographic question asked by Francis Galton, in answer to those who mourned that the great writers and statesmen of the past have so few descendants living today. Galton suspected that even in an increasing population most people will have no descendants, or none beyond two or three generations; most of the increase of the race occurs in relatively few lines of descent. We can say of people in slowly growing populations that either they will have thousands of descendants or they will have none; the chance that they will have just two generation after generation is remote.

In the ordinary branching process it is taken that each individual has a certain probability of generating another individual in each moment, and these probabilities are independent of one another. The parent has the same chance of bearing a child after having born 5 previously as she had at the outset.

What the author has done in this paper is to allow for statistical dependency between siblings. He covers the case where a parent that has had several offspring is less likely to have one more. But it equally covers the case where having had a child shows that the person is fertile, and hence the probability of a further child is raised after the first birth. His results capture the asymptotic growth and fluctuations of such populations, that are followed to their ultimate theoretical condition of stability.

By making its assumptions more realistic, the author has increased the interest of the branching process for students of population.

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