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Speed of coming down from infinity for birth-and-death processes

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  11 January 2017

Vincent Bansaye ,
Sylvie Méléard  and
Mathieu Richard
Vincent Bansaye*
Affiliation:
École Polytechnique
Sylvie Méléard*
Affiliation:
École Polytechnique
Mathieu Richard*
Affiliation:
École Polytechnique
*
* Postal address: CMAP, École Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex, France.
* Postal address: CMAP, École Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex, France.
* Postal address: CMAP, École Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex, France.
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Abstract

We describe in detail the speed of `coming down from infinity' for birth-and-death processes which eventually become extinct. Under general assumptions on the birth-and-death rates, we firstly determine the behavior of the successive hitting times of large integers. We identify two different regimes depending on whether the mean time for the process to go from n+1 to n is negligible or not compared to the mean time to reach n from ∞. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is gradual and a law of large numbers and a central limit theorem for the hitting times sequence hold. By an inversion procedure, we deduce that the process is almost surely equivalent to a nonincreasing function when the time goes to 0. Our results are illustrated by several examples including applications to population dynamics and population genetics. The particular case where the death rate varies regularly is studied in detail.

Keywords

Birth-and-death processescoming down from infinityhitting timescentral limit theorem

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J75: Jump processes 60F15: Strong theorems 60F05: Central limit and other weak theorems
Secondary: 92D25: Population dynamics (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1183 - 1210
Copyright
Copyright © Applied Probability Trust 2017 

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