Geodesic
A~functional limit theorem for a critical branching process in a random environment
Diskretnaya Matematika, Tome 13 (2001) no. 4, pp. 73-91

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Let $\{\xi_n\}$ be a critical branching process in a random environment, and let $m_n$ be the mathematical expectation of $\xi_n$ under the condition that the random environment is fixed. We prove a theorem on convergence of the sequence of branching processes $\{\xi_{[nt]}/m_{[nt]},\ t\in(0,1] \mid \xi_n>0\}$ as $n\to\infty$ in distribution in the corresponding functional space. This theorem extends the earlier result of the author proved under the assumption that the generating function of the number of offspring is linear-fractional. 
@article{DM_2001_13_4_a4,
     author = {V. I. Afanasyev},
     title = {A~functional limit theorem for a critical branching process in a random environment},
     journal = {Diskretnaya Matematika},
     pages = {73--91},
     publisher = {mathdoc},
     volume = {13},
     number = {4},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2001_13_4_a4/}
}
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V. I. Afanasyev. A~functional limit theorem for a critical branching process in a random environment. Diskretnaya Matematika, Tome 13 (2001) no. 4, pp. 73-91. http://geodesic.mathdoc.fr/item/DM_2001_13_4_a4/