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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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