Geodesic
Reduced critical branching processes for small populations
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 795-807

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Let $\left\{ Z(n),n\geq 0\right\} $ be a critical Galton–Watson branching process with finite variance for the offspring number of particles. Assuming that $0$, 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 $ \left\{ Z(m,n),0\leq m\leq n\right\} $, 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. 
@article{TVP_2018_63_4_a8,
     author = {M. Liu and V. A. Vatutin},
     title = {Reduced critical branching processes for small populations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {795--807},
     publisher = {mathdoc},
     volume = {63},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a8/}
}
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M. Liu; V. A. Vatutin. Reduced critical branching processes for small populations. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 795-807. http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a8/