Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T11:39:55.078Z Has data issue: false hasContentIssue false
  • English
  • Français

A connection between the limit and the maximum random variable of a branching process in varying environments

Published online by Cambridge University Press:  14 July 2016

F. C. Klebaner  and
H.-J. Schuh
F. C. Klebaner*
Affiliation:
University of Melbourne
H.-J. Schuh*
Affiliation:
University of Melbourne
*
∗Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, VIC 3052, Australia.
∗Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, VIC 3052, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We show for a certain class of Galton–Watson branching processes in varying environments (Zn)n that moments of the maximum random variable supnZn/Cn exist if and only if the same moments of limnZn/Cn exist, where Cn is a sequence of suitable constants.

Keywords

SUPERCRITICAL GALTON-WATSON BRANCHING PROCESS IN VARYING ENVIRONMENTSEXTINCTION PROBABILITYNORMALIZING CONSTANTSLIMIT RANDOM VARIABLEMAXIMUM RANDOM VARIABLEMOMENTS
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 681 - 684
Copyright
Copyright © Applied Probability Trust 1982 

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

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Cohn, H. (1982) On a property related to convergence in probability and some applications to branching processes. Stoch. Proc. Appl. 12, 59–72.Google Scholar
[3] Cohn, H. and Schuh, H.-J. (1980) On the continuity and the positivity of the finite part of the limit distribution of an irregular branching process with infinite mean. J. Appl. Prob. 17, 696–703.CrossRefGoogle Scholar
[4] Fearn, D. H. (1972) Galton–Watson process with generation dependence. Proc. 6th Berkeley Symp. Math. Statist. Prob. 4, 159–172.Google Scholar
[5] Goettge, R. T. (1975) Limit theorems for the supercritical Galton–Watson process in varying environments. Math. Biosci. 28, 171–190.Google Scholar
[6] Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
[7] Kesten, H. and Stigum, B. P. (1967) Limit theorems for decomposable multidimensional Galton-Watson processes. J. Math. Anal. Appl. 17, 309–338.Google Scholar