Geodesic
On the ratio between the maximal and
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 435-452

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Let $\{\xi_n\}$ be a critical branching process in a random environment. Under some restrictions on the characteristics of the process, we show that the ratio of $\sum_{i=0}^n\xi_i$ to $\max_{0\leqq i\leqq n}\xi_i$, given $\{\xi_n>0\}$, converges in distribution as $n\to\infty$ to a random variable taking on values in $(1,+\infty)$.
Keywords: branching process in a random environment, conditional random walk, conditional limit theorems. 
@article{TVP_2003_48_3_a0,
     author = {V. I. Afanasyev},
     title = {On the ratio between the maximal and},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {435--452},
     publisher = {mathdoc},
     volume = {48},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a0/}
}
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V. I. Afanasyev. On the ratio between the maximal and. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 435-452. http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a0/