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Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  30 March 2016

Kais Hamza  and
Fima C. Klebaner
Kais Hamza
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia. Email address: kais.hamza@monash.edu.
Fima C. Klebaner
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia. Email address: fima.klebaner@monash.edu.
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Abstract

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Looking at a large branching population we determine along which path the population that started at 1 at time 0 ended up in B at time N. The result describes the density process, that is, population numbers divided by the initial number K (where K is assumed to be large). The model considered is that of a Galton-Watson process. It is found that in some cases population paths exhibit the strange feature that population numbers go down and then increase. This phenomenon requires further investigation. The technique uses large deviations, and the rate function based on Cramer's theorem is given. It also involves analysis of existence of solutions of a certain algebraic equation.

Keywords

Branching processlarge deviationsmost likely path

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
Part 3. Biological applications
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 63 - 72
Copyright
Copyright © Applied Probability Trust 2014 

References

Athreya, K. B. (1994). Large deviation rates for branching processes I. Single type case. Ann. Appl. Prob. 4, 779–790.Google Scholar
Athreya, K. B., and Ney, P. E. (2004). Branching Processes. Springer, New York.Google Scholar
Biggins, J. D., and Bingham, N. H. (1993). Large deviations in the supercritical branching process. Adv. Appl. Prob. 25, 757–772.Google Scholar
Dembo, A., and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston, MA.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Kifer, Y. (1990). A discrete-time version of the Wentzell-Freidlin theory. Ann. Prob. 18, 1676–1692.CrossRefGoogle Scholar
Klebaner, F. C., and Zeitouni, O. (1994). The exit problem for a class of density-dependent branching systems. Ann. Appl. Prob. 4, 1188–1205.Google Scholar
Klebaner, F. C., and Liptser, R. (2006). Likely path to extinction in simple branching models with large initial population. J. Appl. Math. Stoch. Anal., 23 pp.Google Scholar
Klebaner, F. C., and Liptser, R. (2009). Large deviations analysis of extinction in branching models. Math. Pop. Studies 15, 55–69.Google Scholar
Ney, P. E., and Vidyashankar, A. N. (2004). Local limit theory and large deviations for supercritical branching processes. Ann. Appl. Prob. 14, 1135–1166.CrossRefGoogle Scholar