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The Time to Extinction of Branching Processes and Log-Convexity: I

Published online by Cambridge University Press:  27 July 2009

M. C. Bhattacharjee
M. C. Bhattacharjee
Affiliation:
Department of StatisticsFlorida State University, Tallahassee, Florida 32306-3033
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Abstract

We show that the time to extinction in critical or subcritical Galton–Watson and Markov branching processes has the antiaging property of log-convex density, and therefore has the decreasing failure rate (DFR) property of reliability theory. Apart from providing new insights into the structure of such extinction time distributions, which cannot generally be expressed in a closed form, a consequence of our result is that one can invoke sharp reliability bounds to provide very simple bounds on the tail and other characteristics of the extinction time distribution. The limit distribution of the residual time to extinction in the subcritical case also follows as a direct consequence. A sequel to this paper will further consider the critical case and other ramifications of the log-convexity of the extinction time distribution.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 1 , Issue 3 , July 1987 , pp. 265 - 278
Copyright
Copyright © Cambridge University Press 1987

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References

1.Agresti, A. (1974). Bounds on the extinction time distribution of a branching process. Adv. in Appl. Probab. 6: 322–335.CrossRefGoogle Scholar
2.Ansmussen, S., & Hering, H. (1983). Branching processes. Boston: Birkhäuser.CrossRefGoogle Scholar
3.Assaf, D., Shaked, M., & Shantikumar, J. G. (1985). First passage times with PFr densities. J. Appl. Probab. 12: 39–46.Google Scholar
4.Attreya, K. B., & Ney, P. E. (1972). Branching processes. New York: Springer-Verlag.CrossRefGoogle Scholar
5.Balkema, A. A., & de Haan, L. (1974). Residual lifetime at great age. Ann. Probab. 2: 792–804.CrossRefGoogle Scholar
6.Barlow, R. E., & Proschan, F. (1965). Mathematical theory of reliability. New York: Wiley.Google Scholar
7.Barlow, R. E., & Proschan, F. (1975). Statistical theory of reliability and life testing: Probability models. New York: Holt, Rinehart, and Winston.Google Scholar
8.Bhattacharjee, M. C. (1985). Tail behavior of age-smooth failure distributions with applications. In Basu, A. P. (ed). Reliability and statistical quality control. Amsterdam: North Holland, pp. 69–86.Google Scholar
9.Heathcote, C. R., & Seneta, E. (1966). Inequalities for branching processes. J. Appl. Probab. 3: 261–267.CrossRefGoogle Scholar
10.Keilson, J. (1979). Markov chain models: Rarity and exponentiality. New York: Springer-Verlag.CrossRefGoogle Scholar
11.Narayan, P. (1981). On the extinction time of a continuous-time Markov branching process. Australian J. Stat. 24(2): 160–164.CrossRefGoogle Scholar
12.Pollak, E. (1969). Bounds for certain branching process. J. Appl. Probab. 6: 201–204.CrossRefGoogle Scholar
13.Pollak, E. (1971). On survival probability and extinction time for some branching processes. J. Appl. Probab. 8: 633–654.CrossRefGoogle Scholar
14.Ross, S. M. (1983). Stochastic processes. New York: Wiley.Google Scholar
15.Seneta, F. (1967). On the transient behaviour of a Poisson branching process. J. Austral. Math. Soc. 7: 465–480.CrossRefGoogle Scholar
16.Seneta, E. (1967). The Galton–Watson process with mean one. J. Appl. Probab. 4: 489–495.CrossRefGoogle Scholar
17.Shaked, M., & Shanthikumar, J. G. (1986). Log-concavity and log-convexity as aging notions. Technical Report, Department of Mathematics, University of Arizona, Tucson, Arizona.Google Scholar
18.Warde, W. D., & Katti, S. K. (1971). Infinite divisibility of discrete distributions II. Ann. Math. Statist 42: 1988–1990.CrossRefGoogle Scholar