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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen and H. Hering, Branching Processes (Birkhäuser, Boston, 1983).
K. B. Athreya and P. E. Ney, Branching Processes (Springer, Berlin, 1972).
N. Berestycki, N. Gantert, P. Mörters, and N. Sidorova,, “Galton-Watson trees with vanishing martingale limit,” arXiv: 1204.3080 [math.PR].
J. D. Biggins and N. H. Bingham, “Large deviations in the supercritical branching process,” Adv. Appl. Probab. 25, 757–772 (1993).
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).
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).
A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions (Cambridge Univ. Press, Cambridge, 2008).
A. De Meyer, “On a theorem of Bingham and Doney,” J. Appl. Probab. 19, 217–220 (1982).
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).
D. Denisov, S. Foss, and D. Korshunov, “Asymptotics of randomly stopped sums in the presence of heavy tails,” Bernoulli 16, 971–994 (2010).
K. Fleischmann and V. Wachtel, “Lower deviation probabilities for supercritical Galton-Watson processes,” Ann. Inst. Henri Poincaré, Probab. Stat. 43, 233–255 (2007).
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).
S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions (Springer, New York, 2011).
T. E. Harris, “Branching processes,” Ann. Math. Stat. 19, 474–494 (1948).
T. E. Harris, The Theory of Branching Processes (Springer, Berlin, 1963).
S. V. Nagaev, “Some limit theorems for large deviations,” Teor. Veroyatn. Primen. 10(2), 231–254 (1965) [Theory Probab. Appl. 10, 214–235 (1965)].
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)].
Author information
Authors and Affiliations
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 282, pp. 288–314.
Rights and permissions
About this article
Cite this article
Wachtel, V.I., Denisov, D.E. & Korshunov, D.A. Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case. Proc. Steklov Inst. Math. 282, 273–297 (2013). https://doi.org/10.1134/S0081543813060205
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543813060205