The Annals of Applied Probability

Branching processes with local dependencies

Peter Olofsson

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A general multitype branching process with sibling dependencies is considered. The dependencies within a group of siblings are described by a joint probability measure, determined by the structure of that particular group. The process is analyzed by means of the embedded macro process, consisting of sibling groups. It is shown that the regular asymptotic behavior of the sibling-dependent process is guaranteed by conditions on the individual reproductions, and that these conditions are exactly the same as those normally required for an ordinary independent process that has the same individual marginals. Convergence results for the expected population size as well as the actual population size are given, and the stable population is described. The sibling-dependent process and the ordinary independent process with the same marginals are compared; some simple examples illustrate the differences and similarities. The results are extended to more general dependencies that are local in the family tree.

Article information

Ann. Appl. Probab., Volume 6, Number 1 (1996), 238-268.

First available in Project Euclid: 18 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F25: $L^p$-limit theorems

General branching process multitype sibling dependencies local dependencies macro process stable population


Olofsson, Peter. Branching processes with local dependencies. Ann. Appl. Probab. 6 (1996), no. 1, 238--268. doi:10.1214/aoap/1034968073.

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