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This article is cited in 6 scientific papers (total in 6 papers)
Tail asymptotics for the supercritical Galton–Watson process in the heavy-tailed case
V. I. Wachtela, D. E. Denisovb, D. A. Korshunovc a Ludwig-Maximilians-Universität München, München, Germany
b University of Manchester, Manchester, UK
c Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Abstract:
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.
Received in November 2012
Citation:
V. I. Wachtel, D. E. Denisov, D. A. Korshunov, “Tail asymptotics for the supercritical Galton–Watson process in the heavy-tailed case”, Branching processes, random walks, and related problems, Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences, Trudy Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 288–314; Proc. Steklov Inst. Math., 282 (2013), 273–297
Linking options:
https://www.mathnet.ru/eng/tm3495https://doi.org/10.1134/S0371968513030205 https://www.mathnet.ru/eng/tm/v282/p288
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