Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

  • V. I. Wachtel
  • D. E. Denisov
  • D. A. Korshunov


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, \(\mathbb{E}e^{\lambda {\rm Z}_1 } = \infty \) for every λ > 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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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • V. I. Wachtel
    • 1
  • D. E. Denisov
    • 2
  • D. A. Korshunov
    • 3
  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany
  2. 2.School of MathematicsUniversity of ManchesterManchesterUK
  3. 3.Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia

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