Geodesic
Tail asymptotics for the supercritical Galton--Watson process in the heavy-tailed case
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 288-314.

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As is well known, for a supercritical Galton–Watson process $Z_n$ whose offspring distribution has mean $m>1$, the ratio $W_n:=Z_n/m^n$ has almost surely a limit, say $W$. We study the tail behaviour of the distributions of $W_n$ and $W$ in the case where $Z_1$ has a heavy-tailed distribution, that is, $\mathbb E\,e^{\lambda Z_1}=\infty$ for every $\lambda>0$. We show how different types of distributions of $Z_1$ lead to different asymptotic behaviour of the tail of $W_n$ and $W$. We describe the most likely way in which large values of the process occur. 
@article{TM_2013_282_a19,
     author = {V. I. Wachtel and D. E. Denisov and D. A. Korshunov},
     title = {Tail asymptotics for the supercritical {Galton--Watson} process in the heavy-tailed case},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {288--314},
     publisher = {mathdoc},
     volume = {282},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_282_a19/}
}
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V. I. Wachtel; D. E. Denisov; D. A. Korshunov. Tail asymptotics for the supercritical Galton--Watson process in the heavy-tailed case. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 288-314. http://geodesic.mathdoc.fr/item/TM_2013_282_a19/

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