Exact distributions of kin numbers in a Galton-Watson process

Abstract

The kin number problem involves the relationship between sibship sizes and offspring numbers, and also numbers of relatives of other degrees of affinity of a random member of a population, to be called Ego. The problem has been well known to demographers for some time, but results obtained only gave expected numbers. Recently a study of it, based on the Galton-Watson process, was made, with a view to obtaining joint distributions (Waugh (1981)). In the latter study it was assumed that the population was large, and thus some of the results obtained were approximations. In the present paper exact distributions are obtained, for any size of population. This can be of use in applications, where the population considered may be a small, isolated tribe or other special group. As a theoretical investigation, it replaces some heuristic arguments with limiting properties that are intrinsic to the process and it makes it possible to evaluate the previous approximations.