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

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

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, \(\mathbb{E}e^{\lambda {\rm Z}_1 } = \infty \) 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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References

  1. 1.
    S. Asmussen and H. Hering, Branching Processes (Birkhäuser, Boston, 1983).CrossRefMATHGoogle Scholar
  2. 2.
    K. B. Athreya and P. E. Ney, Branching Processes (Springer, Berlin, 1972).CrossRefMATHGoogle Scholar
  3. 3.
    N. Berestycki, N. Gantert, P. Mörters, and N. Sidorova,, “Galton-Watson trees with vanishing martingale limit,” arXiv: 1204.3080 [math.PR].Google Scholar
  4. 4.
    J. D. Biggins and N. H. Bingham, “Large deviations in the supercritical branching process,” Adv. Appl. Probab. 25, 757–772 (1993).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    N. H. Bingham and R. A. Doney, “Asymptotic properties of supercritical branching processes. I: The Galton-Watson process,” Adv. Appl. Probab. 6, 711–731 (1974).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    N. H. Bingham and R. A. Doney, “Asymptotic properties of supercritical branching processes. II: Crump-Mode and Jirina processes,” Adv. Appl. Probab. 7, 66–82 (1975).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions (Cambridge Univ. Press, Cambridge, 2008).CrossRefGoogle Scholar
  8. 8.
    A. De Meyer, “On a theorem of Bingham and Doney,” J. Appl. Probab. 19, 217–220 (1982).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    D. Denisov, A. V. Dieker, and V. Shneer, “Large deviations for random walks under subexponentiality: The big-jump domain,” Ann. Probab. 36, 1946–1991 (2008).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    D. Denisov, S. Foss, and D. Korshunov, “Asymptotics of randomly stopped sums in the presence of heavy tails,” Bernoulli 16, 971–994 (2010).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    K. Fleischmann and V. Wachtel, “Lower deviation probabilities for supercritical Galton-Watson processes,” Ann. Inst. Henri Poincaré, Probab. Stat. 43, 233–255 (2007).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    K. Fleischmann and V. Wachtel, “On the left tail asymptotics for the limit law of supercritical Galton-Watson processes in the Böttcher case,” Ann. Inst. Henri Poincaré, Probab. Stat. 45, 201–225 (2009).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions (Springer, New York, 2011).CrossRefMATHGoogle Scholar
  14. 14.
    T. E. Harris, “Branching processes,” Ann. Math. Stat. 19, 474–494 (1948).CrossRefMATHGoogle Scholar
  15. 15.
    T. E. Harris, The Theory of Branching Processes (Springer, Berlin, 1963).CrossRefMATHGoogle Scholar
  16. 16.
    S. V. Nagaev, “Some limit theorems for large deviations,” Teor. Veroyatn. Primen. 10(2), 231–254 (1965) [Theory Probab. Appl. 10, 214–235 (1965)].MathSciNetGoogle Scholar
  17. 17.
    S. V. Nagaev and V. I. Vakhtel, “Probability inequalities for a critical Galton-Watson process,” Teor. Veroyatn. Primen. 50(2), 266–291 (2005) [Theory Probab. Appl. 50, 225–247 (2006)].MathSciNetCrossRefGoogle Scholar

Copyright information

© 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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