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Asymptotic probabilities in a critical age-dependent branching process

Published online by Cambridge University Press:  14 July 2016

Howard J. Weiner
Howard J. Weiner*
Affiliation:
University of California at Davis
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Abstract

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.

The method is also applied to the total progeny in the critical process.

Keywords

CRITICAL BRANCHING PROCESSASYMPTOTIC PROBABILITYAGE-DEPENDENT
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 476 - 485
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Athreya, K. B. and Karlin, S. (1967) Limit theorems for the split times of branching processes. J. Math. Mech. 17, 257–278.Google Scholar
[2] Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
[3] Chover, J. and Ney, P. (1968) The non-linear renewal equation. J. Anal. Math. 21, 381–413.Google Scholar
[4] Weiner, H. (1966) On age-dependent branching processes. J. Appl. Prob. 3, 383–402.Google Scholar