Geodesic
Tail asymptotics for the supercritical Galton–Watson process in the heavy-tailed case
Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 288-314 
V. I. Wachtel; D. E. Denisov; D. A. Korshunov. Tail asymptotics for the supercritical Galton–Watson process in the heavy-tailed case. Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 288-314. http://geodesic.mathdoc.fr/item/TRSPY_2013_282_a19/
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     author = {V. I. Wachtel and D. E. Denisov and D. A. Korshunov},
     title = {Tail asymptotics for the supercritical {Galton{\textendash}Watson} process in the heavy-tailed case},
     journal = {Informatics and Automation},
     pages = {288--314},
     year = {2013},
     volume = {282},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2013_282_a19/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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