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

  • Fabian Kück (a1) and Dominic Schuhmacher (a1)
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.

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Copyright
Corresponding author
* Postal address: Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstraße 7, 37077 Göttingen, Germany.
** Email address: fabian.kueck@mathematik.uni-goettingen.de
*** Email address: schuhmacher@math.uni-goettingen.de
References
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[1]Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
[2]Ba, M., Pardoux, E. and Sow, A. B. (2012). Binary trees, exploration processes, and an extended Ray–Knight theorem. J. Appl. Prob. 49, 210225.
[3]Bailey, N. T. J. (1964). The Elements of Stochastic Processes with Applications to the Natural Sciences. John Wiley, New York.
[4]Gernhard, T. (2008). The conditioned reconstructed process. J. Theoret. Biol. 253, 769778.
[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. 305328.
[6]Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.
[7]Neuts, M. F. and Resnick, S. I. (1971). On the times of births in a linear birthprocess. J. Austral. Math. Soc. 12, 473475.
[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, 228233.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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