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The Markov branching process with density-independent catastrophes. III. The supercritical case

Published online by Cambridge University Press:  24 October 2008

Anthony G. Pakes
Anthony G. Pakes
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia
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Extract

This paper continues the study initiated in [9] of the Markov branching process (MBP) with a density-independent catastrophe component. The population size process (Xt) was investigated in the subcritical and critical cases in [11], and the present paper is concerned with a parallel investigation of the supercritical case. We refer the reader to [11] for definitions and notation, and we will not repeat them here.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 1 , January 1990 , pp. 177 - 192
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Asmussen, S. and Hering, H.. Branching Processes (Birkhäuser, 1983).Google Scholar
[2]Barbour, A. D.. The asymptotic behaviour of birth and death and some related processes. Adv. in Appl. Probab. 7 (1975), 28–43.CrossRefGoogle Scholar
[3]Erickson, K. B.. Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 (1970), 263–291.CrossRefGoogle Scholar
[4]Feller, W.. An Introduction to Probability Theory and its Applications, vol. 2, 2nd ed. (Wiley, 1971).Google Scholar
[5]Flaspohler, D. C.. Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26 (1974), 351–356.CrossRefGoogle Scholar
[6]Grey, D. R.. Supercritical branching processes with density independent catastrophes. Math. Proc. Cambridge Philos. Soc. 104 (1988), 413–416.Google Scholar
[7]Pakes, A. G.. The Markov branching-catastrophe process. Stochastic Process. Appl. 23 (1986), 1–33.Google Scholar
[8]Pakes, A. G.. Limit theorems for the population size of a birth and death process allowing catastrophes. J. Math. Biol. 25 (1987), 307–325.CrossRefGoogle ScholarPubMed
[9]Pakes, A. G.. The Markov branching process with density-independent catastrophes I. Behaviour of extinction probabilities. Math. Proc. Cambridge Philos. Soc. 103 (1988), 351–366.Google Scholar
[10]Pakes, A. G.. The supercritical birth, death and catastrophe process: Limit theorems on the set of non-extinction. J. Math. Biol. 26 (1988), 405–420.Google Scholar
[11]Pakes, A. G.. The Markov branching process with density-independent catastrophes II. The subcritical and critical cases. Math. Proc. Cambridge Philos. Soc. 106 (1989), 369–383.CrossRefGoogle Scholar
[12]Pakes, A. G.. Asymptotic results for the extinction time of Markov branching processes allowing emigration. I. Random walk decrements. Adv. in Appl. Prob. (to appear).Google Scholar
[13]Pakes, A. G.. A complementary note on the supercritical birth, death and catastrophe process. J. Math. Biol. (to appear).Google Scholar
[14]Pakes, A. G. and Pollett, P.. The supercritical birth, death and catastrophe process: Limit theorems on the set of non-extinction. Stochastic Process. Appl. (to appear).Google Scholar