Geodesic
A critical Galton–Watson branching process with emigration
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 482-497 
V. A. Vatutin. A critical Galton–Watson branching process with emigration. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 482-497. http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a2/
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     author = {V. A. Vatutin},
     title = {A critical {Galton{\textendash}Watson} branching process with emigration},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {482--497},
     year = {1977},
     volume = {22},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper, an example of critical $\varphi$-branching processes introduced in [1] is investigated. For $\varphi(n)=\max\{0,n-1\}$, we derive an asymptotic formula for the probability $\mathbf P\{\mu(t)>0\mid\mu(0)=m\ge 2\}$ as $t\to\infty$. Here $\mu(t)$ is the number of particles at time $t$. We also obtain a conditional limit theorem for this process which is analogous to a well-known result for a critical Galton–Watson process.