Geodesic
Limit theorems for a moderately subcritical branching process in a random environment
Diskretnaya Matematika, Tome 10 (1998) no. 1, pp. 141-157 
V. I. Afanasyev. Limit theorems for a moderately subcritical branching process in a random environment. Diskretnaya Matematika, Tome 10 (1998) no. 1, pp. 141-157. http://geodesic.mathdoc.fr/item/DM_1998_10_1_a12/
@article{DM_1998_10_1_a12,
     author = {V. I. Afanasyev},
     title = {Limit theorems for a moderately subcritical branching process in a random environment},
     journal = {Diskretnaya Matematika},
     pages = {141--157},
     year = {1998},
     volume = {10},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1998_10_1_a12/}
}
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Let $\{\xi_n\}$ be a moderately subcritical branching process in a random environment with linear-fractional generating functions, $m_n$ be the conditional expectation of $\xi_n$ with respect to the random environment. We prove theorems on convergence of the sequence of random processes $$ \{\xi_{[nt]}/m_{[nt]},\,t\in(0,1)\mid \xi_n>0\} $$ as $n\to\infty$ in distribution, and of the initial and final segments of the random sequence $\xi_0/m_0,\xi_1/m_1,\ldots,\xi_n/m_n$ considered under the condition that $\{\xi_n>0\}$.