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On the times of births in a linear birthprocess

Published online by Cambridge University Press:  09 April 2009

Marcel F. Neuts
Affiliation:
Cornell University
Sidney I. Resnick
Affiliation:
Purdue University
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Abstract

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If S1, S2,… are the times of successive births in a pure birth-process with linear birthrate and i > 0 individuals initially, then given that there are k > 0 births in (0, t), the random variables eλS1—1, eλS2−1,λS1−1 are distributed (conditionally) like the order statistics of k independent random variables, all uniformly distributed on the interval (0, eλt–1), regardless of the initial population size. A similar property holds for the pure death process.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Karlin, Samuel, A First Course in Stochastic Processes (Academic Press, New York and London, 1966).Google Scholar