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 Tr. Mat. Inst. Steklova, 2013, Volume 282, Pages 288–314 (Mi tm3495)

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.

DOI: https://doi.org/10.1134/S0371968513030205

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 282, 273–297

Bibliographic databases:

Document Type: Article
UDC: 519.218.23

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, Tr. Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 288–314; Proc. Steklov Inst. Math., 282 (2013), 273–297

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3495
• https://doi.org/10.1134/S0371968513030205
• http://mi.mathnet.ru/eng/tm/v282/p288

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This publication is cited in the following articles:
1. N. Berestycki, N. Gantert, P. Mörters, N. Sidorova, “Galton-Watson trees with vanishing martingale limit”, J. Stat. Phys., 155:4 (2014), 737–762
2. S. V. Nagaev, “Probability inequalities for Galton–Watson processes”, Theory Probab. Appl., 59:4 (2015), 611–640
3. M. Barczy, Z. Bosze, G. Pap, “Regularly varying non-stationary Galton-Watson processes with immigration”, Stat. Probab. Lett., 140 (2018), 106–114
4. Abraham R., Delmas J.-F., “Asymptotic Properties of Expansive Galton-Watson Trees”, Electron. J. Probab., 24 (2019), 15
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