Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T18:19:37.179Z Has data issue: false hasContentIssue false
  • English
  • Français

On distribution tails and expectations of maxima in critical branching processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

K. A. Borovkov  and
V A. Vatutin
K. A. Borovkov*
Affiliation:
Steklov Mathematical Institute
V A. Vatutin*
Affiliation:
Steklov Mathematical Institute
*
∗Postal address for both authors: Steklov Mathematical Institute, Vavilov st. 42, 117966 Moscow GSP-1, Russia.
∗Postal address for both authors: Steklov Mathematical Institute, Vavilov st. 42, 117966 Moscow GSP-1, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.

Keywords

BRANCHING PROCESSMAXIMA OF GALTON–WATSON PROCESSESINFINITE VARIANCEGENERAL BRANCHING PROCESS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 614 - 622
Copyright
Copyright © Applied Probability Trust 1996 

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.)

Footnotes

Research supported by the International Science Foundation Grant MQP000.

V. A. Vatutin was partially supported by the Russian Fundamental Science Foundation Grant no. 93-01–01443.

References

Athreya, K. B. (1988) On the maximum sequence in a critical branching process. Ann. Prob. 16, 502–507.Google Scholar
Bahr, B. Von and Esseen, C.-G. (1965) Inequalities for the r-th moment of a sum of random variables, 1 r 2. Ann. Math. Statist. 36, 299–303.CrossRefGoogle Scholar
Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer, New York.CrossRefGoogle Scholar
Borovkov, K. A. (1985) On the convergence of branching processes to a diffusion process. Theory Prob. Appl. 30, 496–506.Google Scholar
Borovkov, K. A. (1988) A method of proving limit theorems for branching processes. Theory Prob. Appl. 33, 105–113.Google Scholar
Doney, R. A. (1983) A note on conditioned random walks. J. Appl. Prob. 20, 409–412.Google Scholar
Dwass, M. (1969) The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682–686.CrossRefGoogle Scholar
Dwass, M. (1975) Branching processes in simple random walk. Proc. Amer. Math. Soc. 51, 270–274.Google Scholar
Kämmerle, K. and Schuh, H.-J. (1986) The maximum in critical Galton-Watson and birth and death processes. J. Appl. Prob. 23, 601–613.CrossRefGoogle Scholar
Lindvall, T. (1976) On the maximum of a branching process. Scand. J. Statist. Theory Appl. 3, 209–214.Google Scholar
Pakes, A. G. (1978) On the maximum and absorption time of a left-continuous random walk. J. Appl. Prob. 15, 292–299.Google Scholar
Pakes, A. G. (1987) Remarks on the maxima of a martingale sequence with applications to the simple critical branching process. J. Appl. Prob. 24, 768–772.Google Scholar
Slack, R. S. (1968) A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, Princeton, NJ.Google Scholar
Topchii, V. A. (1987) Properties of the non-extinction probability of Crump-Mode-Jagers branching processes under mild constraints. Sib. Math. J. 28, 178–192. (In Russian.)Google Scholar
Viskov, O. V. (1970) Some comments on branching processes. Math. Zametki 8, 701–705.Google Scholar
Weiner, H. (1984) Moments of the maximum in a critical branching process. J. Appl. Prob. 21, 920–923.Google Scholar