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Regeneration of branching processes with immigration in varying environments

Part of: Markov processes

Published online by Cambridge University Press:  21 November 2024

Hong-Yan Sun ,
Hua-Ming Wang ,
Bao-Zhi Li  and
Hui Yang
Hong-Yan Sun*
Affiliation:
China University of Geosciences
Hua-Ming Wang*
Affiliation:
Anhui Normal University
Bao-Zhi Li*
Affiliation:
Anhui Normal University
Hui Yang*
Affiliation:
Minzu University of China
*
*Postal address: School of Sciences, China University of Geosciences, Beijing 100083, China. Email: [email protected]
**Postal address: School of Mathematics and Statistics, Anhui Normal University, Wuhu 241003, China.
**Postal address: School of Mathematics and Statistics, Anhui Normal University, Wuhu 241003, China.
***Postal address: School of Sciences, Minzu University of China, Beijing 100081, China.
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Abstract

We consider linear-fractional branching processes (one-type and two-type) with immigration in varying environments. For $n\ge0$, let $Z_n$ count the number of individuals of the nth generation, which excludes the immigrant who enters the system at time n. We call n a regeneration time if $Z_n=0$. For both the one-type and two-type cases, we give criteria for the finiteness or infiniteness of the number of regeneration times. We then construct some concrete examples to exhibit the strange phenomena caused by the so-called varying environments. For example, it may happen that the process is extinct, but there are only finitely many regeneration times. We also study the asymptotics of the number of regeneration times of the model in the example.

Keywords

Branching processesimmigrationregeneration timescontinued fractionsproduct of nonnegative matrices

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 422 - 444
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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