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On the age of a randomly picked individual in a linear birth-and-death process

Part of: Distribution theory Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  28 March 2018

Fabian Kück  and
Dominic Schuhmacher
Fabian Kück*
Affiliation:
University of Göttingen
Dominic Schuhmacher*
Affiliation:
University of Göttingen
*
* Postal address: Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstraße 7, 37077 Göttingen, Germany.
* Postal address: Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstraße 7, 37077 Göttingen, Germany.
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Abstract

We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.

Keywords

Age distributionbirth-and-death processcontour process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60E05: Distributions 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 82 - 93
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York. Google Scholar
[2]Ba, M., Pardoux, E. and Sow, A. B. (2012). Binary trees, exploration processes, and an extended Ray–Knight theorem. J. Appl. Prob. 49, 210–225. CrossRefGoogle Scholar
[3]Bailey, N. T. J. (1964). The Elements of Stochastic Processes with Applications to the Natural Sciences. John Wiley, New York. Google Scholar
[4]Gernhard, T. (2008). The conditioned reconstructed process. J. Theoret. Biol. 253, 769–778. Google Scholar
[5]Harris, T. E. (1951). Some mathematical models for branching processes. In Proc. Second Berkeley Symp. Math. Statist. Prob., University of California Press, Berkeley, CA, pp. 305–328. Google Scholar
[6]Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London. Google Scholar
[7]Neuts, M. F. and Resnick, S. I. (1971). On the times of births in a linear birthprocess. J. Austral. Math. Soc. 12, 473–475. Google Scholar
[8]Stadler, T., Kühnert, D., Bonhoeffer, S. and Drummond, A. J. (2013). Birth–death skyline plot reveals temporal changes of epidemic spread in HIV and hepatitis C virus (HCV). Proc. Nat. Acad. Sci. 110, 228–233. Google Scholar