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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.
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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 

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Footnotes

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

References

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