Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-23T05:07:21.478Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for the simple branching process allowing immigration, II. The case of infinite offspring mean

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour  and
Anthony G. Pakes
A. D. Barbour*
Affiliation:
University of Cambridge
Anthony G. Pakes*
Affiliation:
Monash University
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, U.K.
∗∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents some limit theorems for the simple branching process allowing immigration, {Xn}, when the offspring mean is infinite. It is shown that there exists a function U such that {e–nU/(Xn)} converges almost surely, and if s = ∑ bj, log+U(j) < ∞, where {bj} is the immigration distribution, the limit is non-defective and non-degenerate but is infinite if s = ∞.

When s = ∞, limit theorems are found for {U(Xn)} which involve a slowly varying non-linear norming.

Keywords

BIENAYMÉ–GALTON–WATSON BRANCHING PROCESSIMMIGRATIONMARTINGALE CONVERGENCELIMIT THEOREMSREGULAR VARIATION
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 1 , March 1979 , pp. 63 - 72
Copyright
Copyright © Applied Probability Trust 1979 

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 sponsored in part by O.N.R. contract N00014-75-C-0453, awarded to the Department of Statistics, Princeton University.

References

1. Hudson, I. L. and Seneta, E. (1977) The simple branching process with infinite mean. J. Appl. Prob. 14, 836–842.Google Scholar
2. Pakes, A. G. (1976) Some limit theorems for a supercritical branching process allowing immigration. J. Appl. Prob. 13, 17–26.CrossRefGoogle Scholar
3. Pakes, A. G. (1979) Limit theorems for the simple branching process allowing immigration I. The case of finite offspring mean. Adv. Appl. Prob. 11, 31–62.Google Scholar
4. Rényi, A. (1958) On mixing sequences of sets. Acta. Math. Acad. Sci. Hungar. 9, 215–228.Google Scholar
5. Rényi, A. and Révész, P. (1958) On mixing sequences of random variables. Acta Math. Acad. Sci. Hungar. 9, 389–393.Google Scholar
6. Schuh, H.-J. and Barbour, A. D. (1977) Normalising constants for the branching process with infinite mean. Adv. Appl. Prob. 9, 681–723.CrossRefGoogle Scholar
7. Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar