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

Part of: Markov processes Distribution theory - Probability Distribution theory

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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