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Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 4, Pages 795–807 (Mi tvp5235)  

Short Communications

Reduced critical branching processes for small populations

M. Liua, V. A. Vatutinbc

a School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, China
b Beijing Normal University, Beijing, China
c Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Let $\{ Z(n),n\geq 0\} $ be a critical Galton–Watson branching process with finite variance for the offspring number of particles. Assuming that $0<Z(n)\leq \varphi (n)$, where either $\varphi (n)=an$ for some $a>0$ or $\varphi (n)=o(n)$ as $n\rightarrow \infty $, we study the structure of the process $\{ Z(m,n),0\leq m\leq n\} $, where $Z(m,n)$ is the number of particles in the initial process at moment $m\leq n$ having a positive number of descendants at moment $n$.

Keywords: critical branching process, reduced processes, conditional limit theorems.

Funding Agency Grant Number
National Natural Science Foundation of China 11531001
11626245
High-end Foreign Experts Program GDW20171100029


DOI: https://doi.org/10.4213/tvp5235

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English version:
Theory of Probability and its Applications, 2019, 63:4, 648–656

Bibliographic databases:

Document Type: Article
Received: 18.06.2018
Revised: 25.06.2018
Accepted:26.06.2018

Citation: M. Liu, V. A. Vatutin, “Reduced critical branching processes for small populations”, Teor. Veroyatnost. i Primenen., 63:4 (2018), 795–807; Theory Probab. Appl., 63:4 (2019), 648–656

Citation in format AMSBIB
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