The Annals of Applied Probability

Branching processes with local dependencies

Peter Olofsson

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

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

References

• ASH, R. B. 1972. Real Analy sis and Probability. Academic Press, London. Z.
• BROBERG, P. 1987. Sibling dependencies in branching populations. Ph.D. dissertation, Goteborg ¨ Univ. Z.
• CRUMP, K. S. and MODE, C. J. 1969. An age-dependent branching process with correlations among sister cells. J. Appl. Probab. 6 205 210.
• JAGERS, P. 1989. General branching processes as Markov fields. Stochastic Process. Appl. 32 183 212. Z.
• JAGERS, P. 1992. Stabilities and instabilities in population dy namics. J. Appl. Probab. 29 770 780. Z.
• JAGERS, P. 1995. Branching processes as population dy namics. Bernoulli 1 191 200. Z.
• JAGERS, P. and NERMAN, O. 1992. The asy mptotic composition of supercritical multi-ty pe branching populations. Technical report, Dept. Mathematics, Chalmers Univ. Technology and Goteborg Univ. ¨ Z.
• KUBITSCHEK, H. E. 1967. Cell generation times: ancestral and internal controls. Proc. Fifth Berkeley Sy mp. Math. Statist. Probab. 4 549 572. Univ. California Press, Berkeley. Z.
• NERMAN, O. 1984. The growth and composition of supercritical branching populations on general ty pe spaces. Technical report, Dept. Mathematics, Chalmers Univ. Technology and Goteborg Univ. ¨ Z.
• NERMAN, O. and JAGERS, P. 1984. The stable doubly infinite pedigree process of supercritical branching populations. Z. Wahrsch. Verw. Gebiete 65 445 460. Z.
• OLOFSSON, P. 1994. General branching processes with local dependencies. Ph.D. dissertation, Goteborg Univ. ¨Z.
• POWELL, E. 1955. Some features of the generation times of individual bacteria. Biometrika 42 16 44. Z.
• SHURENKOV, V. M. 1989. Ergodicheskie Protsessy Markova. Nauka, Moscow. Z.
• SHURENKOV, V. M. 1992. Markov renewal theory and its application to Markov ergodic processes. Technical report, Dept. Mathematics, Chalmers Univ. Technology and Goteborg ¨ Univ.