Geodesic
Maximum of the critical Galton--Watson processes and left-continuous random walks
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 1, pp. 21-34

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Let $Z(n)$, $n=0,1,\dots$ be a critical Galton–Watson branching process, $Z(0)=1$. Under mild conditions on the distribution of $Z(1)$, we prove that $$ \mathsf{E}\max_{1\le k\le n}Z(k)\sim\log n, \qquad n\to\infty. $$
Keywords: critical branching process, maximum of a branching process, the von Bahr–Esseen inequality, left-continuous random walk. 
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     author = {V. A. Vatutin and V. A. Topchii},
     title = {Maximum of the critical {Galton--Watson} processes and left-continuous random walks},
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     year = {1997},
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V. A. Vatutin; V. A. Topchii. Maximum of the critical Galton--Watson processes and left-continuous random walks. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 1, pp. 21-34. http://geodesic.mathdoc.fr/item/TVP_1997_42_1_a1/