Geodesic
Sevastyanov branching processes with non-homogeneous Poisson immigration
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 181-194

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Sevastyanov age-dependent branching processes allowing an immigration component are considered in the case when the moments of immigration form a non-homogeneous Poisson process with intensity $r(t)$. The asymptotic behavior of the expectation and of the probability of non-extinction is investigated in the critical case depending on the asymptotic rate of $r(t)$. Corresponding limit theorems are also proved using different types of normalization. Among them we obtained limiting distributions similar to the classical ones of Yaglom (1947) and Sevastyanov (1957) and also discovered new phenomena due to the non-homogeneity. 
@article{TM_2013_282_a14,
     author = {Kosto V. Mitov and Nikolay M. Yanev},
     title = {Sevastyanov branching processes with non-homogeneous {Poisson} immigration},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {181--194},
     publisher = {mathdoc},
     volume = {282},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_282_a14/}
}
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Kosto V. Mitov; Nikolay M. Yanev. Sevastyanov branching processes with non-homogeneous Poisson immigration. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 181-194. http://geodesic.mathdoc.fr/item/TM_2013_282_a14/