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Stationary measures and the continuous-state branching process conditioned on extinction

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  07 October 2024

Rongli Liu ,
Yan-Xia Ren  and
Ting Yang
Rongli Liu*
Affiliation:
Beijing Jiaotong University
Yan-Xia Ren*
Affiliation:
Peking University
Ting Yang*
Affiliation:
Beijing Institute of Technology
*
*Postal address: School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, PR China. Email address: rlliu@bjtu.edu.cn
**Postal address: LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University, Beijing 100871, PR China. Email address: yxren@math.pku.edu.cn
***Postal address: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China. Email address: yangt@bit.edu.cn
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Abstract

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

Keywords

Continuous-state branching processstationary measurevague convergenceconditional limit theoremssize-biased measure

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 2 , June 2025 , pp. 576 - 602
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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