Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T18:53:38.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the maxima of a martingale sequence with application to the simple critical branching process

Published online by Cambridge University Press:  14 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let where {Zn, ℱn} is a non-negative submartingale satisfying Ei(Zn log Zn) →∞. It is shown that When {Zn} is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

Keywords

MARTINGALE INEQUALITIES
Type
Short Communications
Information
Journal of Applied Probability , Volume 24 , Issue 3 , September 1987 , pp. 768 - 772
Copyright
Copyright © Applied Probability Trust 1987 

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 carried out while the author was visiting Colorado State University, Fort Collins.

References

[1] Athreya, K. B. (1986) On the maximum sequence in a critical branching process. Iowa State University Technical Report.Google Scholar
[2] Athreya, K. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[3] Chow, Y. S. and Teicher, H. (1978) Probability Theory. Springer-Verlag, New York.Google Scholar
[4] Kammerle, K. and Schuh, H.-J. (1986) The maximum in critical Galton–Watson and birth and death processes. J. Appl. Prob. 23, 601–613.Google Scholar
[5] Lindvall, T. (1976) On the maximum of a branching process. Scand J. Statist. 3, 204–214.Google Scholar
[6] Pakes, A. G. (1979) Limit theorems for the simple branching process with immigration, I. The case of finite offspring mean. Adv. Appl. Prob. 11, 31–62.CrossRefGoogle Scholar
[7] Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
[8] Weiner, H. (1984) Moments of the maximum in a critical branching process. J. Appl. Prob. 21, 920–923.Google Scholar