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Limit theorems for a branching process with disasters

Published online by Cambridge University Press:  14 July 2016

K. B. Athreya  and
N. Kaplan
K. B. Athreya
Affiliation:
Institute of Mathematical Statistics, University of Copenhagen
N. Kaplan
Affiliation:
Institute of Mathematical Statistics, University of Copenhagen
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Abstract

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

Keywords

AGE-DEPENDENT BRANCHING PROCESSAGE-DISTRIBUTION DISASTERSSUPERCRITICAL
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 466 - 475
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Athreya, K. B. and Kaplan, N. (1976) Convergence of the age distribution in the one-dimensional supercritical age dependent branching process. Ann. Prob., To appear.Google Scholar
[2] Harris, T. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
[3] Kaplan, N., Sudbury, A. and Nilsen, T. S. (1975) A branching process with disasters. J. Appl. Prob. 12, 130–134.Google Scholar
[4] Hewitt, E. and Stromberg, K. (1965) Real and Abstract Analysis. Springer, Berlin.Google Scholar