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December 2016 Speed of coming down from infinity for birth-and-death processes
Vincent Bansaye, Sylvie Méléard, Richard Mathieu
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Adv. in Appl. Probab. 48(4): 1183-1210 (December 2016).

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.

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Vincent Bansaye. Sylvie Méléard. Richard Mathieu. "Speed of coming down from infinity for birth-and-death processes." Adv. in Appl. Probab. 48 (4) 1183 - 1210, December 2016.

Information

Published: December 2016
First available in Project Euclid: 24 December 2016

zbMATH: 1358.60087
MathSciNet: MR3595771

Subjects:
Primary: 60F05, 60F15, 60J27, 60J75
Secondary: 92D25

Rights: Copyright © 2016 Applied Probability Trust

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Vol.48 • No. 4 • December 2016
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