Geodesic
A functional limit theorem for the logarithm of a moderately subcritical branching process in a random environment
Diskretnaya Matematika, Tome 10 (1998) no. 3, pp. 131-147
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Let $\{\xi_n\}$ be a moderately subcritical branching process in a random environment with linear-fractional generating functions. We prove that, as $n\to\infty$, the sequence of stochastic processes $\{\ln\xi_{[nt]}/(\Delta \sqrt n),\ t\in [0,1]\mid \xi_n>0\}$, where $\Delta$ is some positive constant, converges in distribution to the Brownian excursion $\{W_0^+(t),\ t\in [0,1]\}$ in the space $D[0,1]$ with Skorokhod topology. 
@article{DM_1998_10_3_a10,
     author = {V. I. Afanasyev},
     title = {A functional limit theorem for the logarithm of a moderately subcritical branching process in a random environment},
     journal = {Diskretnaya Matematika},
     pages = {131--147},
     year = {1998},
     volume = {10},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1998_10_3_a10/}
}
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V. I. Afanasyev. A functional limit theorem for the logarithm of a moderately subcritical branching process in a random environment. Diskretnaya Matematika, Tome 10 (1998) no. 3, pp. 131-147. http://geodesic.mathdoc.fr/item/DM_1998_10_3_a10/