Speed of coming down from infinity for birth-and-death processes. (English) Zbl 1358.60087

Bansaye, Vincent; Méléard, Sylvie; Richard, Mathieu

Adv. Appl. Probab. 48, No. 4, 1183-1210 (2016).

Summary: 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 \(\infty\). 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.

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J27	Continuous-time Markov processes on discrete state spaces
60J75	Jump processes (MSC2010)
60F05	Central limit and other weak theorems
60F15	Strong limit theorems
60J85	Applications of branching processes
92D25	Population dynamics (general)

Keywords:

birth-and-death processes; hitting times; central limit theorem; population dynamics; population genetics