Geodesic
On the maximum of a critical branching process in a random environment
Diskretnaya Matematika, Tome 11 (1999) no. 2, pp. 86-102 
V. I. Afanasyev. On the maximum of a critical branching process in a random environment. Diskretnaya Matematika, Tome 11 (1999) no. 2, pp. 86-102. http://geodesic.mathdoc.fr/item/DM_1999_11_2_a4/
@article{DM_1999_11_2_a4,
     author = {V. I. Afanasyev},
     title = {On the maximum of a critical branching process in a random environment},
     journal = {Diskretnaya Matematika},
     pages = {86--102},
     year = {1999},
     volume = {11},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1999_11_2_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\{\xi_n\}$ be a critical branching process in a random environment with linear-fractional generating functions. We demonstrate that, under some conditions, as $x\to\infty$, $$ \mathsf P\Bigl(\sup_n\xi_n>x\Bigr)\sim \frac{c_0}{\ln x},\qquad \mathsf P\biggl(\sum_{n=0}^\infty\xi_n>x\biggr)\sim \frac{c_0}{\ln x}, $$ where $c_0$ is a positive constant.