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

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
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Abstract

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process.

The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

Keywords

BIENAYMÉ–GALTON–WATSONIMMIGRATIONLIMIT THEOREMNON-LINEAR NORMINGMARKOV BRANCHING PROCESSSLOW VARIATIONSTATIONARY MEASUREREVERSED TIME
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 1 , March 1979 , pp. 31 - 62
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research carried out at Princeton University and partially supported by O.N.R. contract N00014-75-C-0453. The author thanks Geof Watson and Peter Bloomfield for their hospitality.

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