Journal of Applied Probability

On largest offspring in a critical branching process with finite variance

Jean Bertoin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail with index −α for α>2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

Article information

Source
J. Appl. Probab., Volume 50, Number 3 (2013), 791-800.

Dates
First available in Project Euclid: 5 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1378401236

Digital Object Identifier
doi:10.1239/jap/1378401236

Mathematical Reviews number (MathSciNet)
MR3188595

Zentralblatt MATH identifier
1277.60045

Subjects
Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching process maximal offspring extreme value theory Cox process

Citation

Bertoin, Jean. On largest offspring in a critical branching process with finite variance. J. Appl. Probab. 50 (2013), no. 3, 791--800. doi:10.1239/jap/1378401236. https://projecteuclid.org/euclid.jap/1378401236


Export citation

References

  • Athreya, K. B. (1988). On the maximum sequence in a critical branching process. Ann. Prob. 16, 502–507.
  • Bertoin, J. (2011). On the maximal offspring in a critical branching process with infinite variance. J. Appl. Prob. 48, 576–582.
  • Borovkov, K. A. and Vatutin, V. A. (1996). On distribution tails and expectations of maxima in critical branching processes. J. Appl. Prob. 33, 614–622.
  • Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682–686.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
  • Lindvall, T. (1976). On the maximum of a branching process. Scand. J. Statist. 3, 209–214.
  • Pakes, A. G. (1998). Extreme order statistics on Galton-Watson trees. Metrika 47, 95–117.
  • Pitman, J. (2006). Combinatorial Stochastic Processes. (Lecture Notes Math. 1875), Springer, Berlin.
  • Rahimov, I. and Yanev, G. P. (1999). On maximum family size in branching processes. J. Appl. Prob. 36, 632–643.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Vatutin, V. A., Wachtel, V. and Fleischmann, K. (2008). Critical Galton-Watson branching processes: the maximum of the total number of particles within a large window. Theory Prob. Appl. 52, 470–492.
  • Yanev, G. P. (2007). Revisiting offspring maxima in branching processes. Pliska Stud. Math. Bulgar. 18, 401–426.
  • Yanev, G. P. (2008). A review of offspring extremes in branching processes. In Records and Branching Processes, eds M. Ahsanullah and G. P. Yanev, Nova Science, pp. 127–145.