## The Annals of Applied Probability

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

### Branching processes with local dependencies

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 18 October 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1034968073

**Digital Object Identifier**

doi:10.1214/aoap/1034968073

**Mathematical Reviews number (MathSciNet)**

MR1389839

**Zentralblatt MATH identifier**

0863.60080

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 60F25: $L^p$-limit theorems

**Keywords**

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

#### Citation

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