Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-12T14:29:33.798Z Has data issue: false hasContentIssue false
  • English
  • Français

On maximum family size in branching processes

Part of: Markov processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Ibrahim Rahimov  and
George P. Yanev
Ibrahim Rahimov*
Affiliation:
Uzbek Academy of Sciences
George P. Yanev*
Affiliation:
Bulgarian Academy of Sciences
*
∗Postal address: Box 1339, King Fahd University of Petroleum Minerals, Dhahran 31261, Saudi Arabia.
∗∗Postal address: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Box 373, 1090 Sofia, Bulgaria. Email address: gyanev@tarski.math.usf.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

Keywords

Bienaymé–Galton–Watson branching processmax-stabilitymax-semi-stabilityrandom sample sizetransfer theorems

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 632 - 643
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, B. C. and Villaseñor, J. A. (1996). The tallest man in the world. In Statistical Theory and Applications: Papers in Honor of Herbert A. David, ed. Nagaraja, H. N., Sen, P. K. and Morrison, D. F. Springer, Berlin, pp. 81–88.CrossRefGoogle Scholar
Bingham, N. H., and Doney, R. A. (1974). Asymptotic properties of super-critical branching processes. I: Galton–Watson process. Adv. Appl. Prob. 6, 711–731.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation (Encyclopedia of Mathematics and its Applications, Vol. 27). CUP, Cambridge.Google Scholar
Bojanic, R., and Seneta, E. (1971). Slowly varying functions and asymptotic relations. J. Math. Anal. Appl. 34, 302–315.Google Scholar
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.Google Scholar
Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Krieger, Melbourne, FL.Google Scholar
Gnedenko, B. V., and Gnedenko, D. B. (1982). On Laplace and logistical distributions as limit ones in the probability thoery. Serdica, Bulgarian Math. J. 8, 229–234. (In Russian.)Google Scholar
Grinevich, I. V. (1993). Domains of attraction of the max-semistable laws under linear and power normalizations. Theory Prob. Appl. 38, 640–650.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, London.Google Scholar
Jagers, P., and Nerman, O. (1984). Limit theorems for sums determined by branching and other exponentially growing processes. Stoch. Proc. Appl. 17, 47–71.Google Scholar
Lamperti, J. (1972). Remarks on maximal branching processes. Theory Prob. Appl. 17, 44–53.Google Scholar
Leadbetter, M. R., Lindgren, G., and Rootzen, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin.Google Scholar
Rahimov, I. (1995). Random Sums and Branching Processes (Lecture Notes in Statist. 96). Springer, Berlin.Google Scholar
Rahimov, I., and Yanev, G. (1996). On a maximal sequence associated with simple branching processes. Preprint No. 6, Institute of Mathematics and Informatics, Sofia, Bulgaria.Google Scholar
Resnick, S. (1987). Extreme Value Distributions, Regular Variations, and Point Processes. Springer, Berlin.Google Scholar
Wilms, R. (1994). Fractional parts of random variables. , Technical University of Eindhoven, Eindhoven, Holland.Google Scholar