Geodesic
Asymptotic behaviour of the non-extinction probability for a~critical branching process
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 143-149

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Critical age-dependent branching process with $k$ types $N_1,N_2,\dots,N_k$ of particles are considered. We suppose that particles' reproduction power depends on their age. Let $z_j^i(t)$ be the number of particles of type $N_j$ at time $t$ given that at time $t=0$ there was only one particle of type $N_i$. We derive an asymptotic formula for the probability $\mathbf P\{z_1^i(t)+z_2^i(t)+\dots+z_k^i(t)>0\}$ as $t\to\infty$. The result obtained is analogous to that of Goldstein [2]. 
@article{TVP_1977_22_1_a12,
     author = {V. A. Vatutin},
     title = {Asymptotic behaviour of the non-extinction probability for a~critical branching process},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {143--149},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a12/}
}
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V. A. Vatutin. Asymptotic behaviour of the non-extinction probability for a~critical branching process. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 143-149. http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a12/