Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-11T01:57:47.047Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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 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.

References

1. Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
2. Barbour, A. D. and Pakes, A. G. (1979) Limit theorems for the simple branching process allowing immigration, II. The case of infinite offspring mean. Adv. Appl. Prob. 11,CrossRefGoogle Scholar
3. Cohn, H. (1977) On the convergence of the supercritical branching process with immigration. J. Appl. Prob. 14, 387–390.CrossRefGoogle Scholar
4. Cohn, H. (1977) Almost sure convergence of branching processes. Z. Wahrscheinlichkeitsth. 38, 73–81.Google Scholar
5. Esty, W. W. (1975) The reversed Galton–Watson process. J. Appl. Prob. 12, 574–580.CrossRefGoogle Scholar
6. Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
7. Foster, J. (1969) Branching Processes Involving Immigration. , University of Wisconsin.Google Scholar
8. Foster, J. and Williamson, J. A. (1971) Limit theorems for the Galton–Watson process with time-dependent immigration. Z. Wahrscheinlichkeitsth. 20, 227–235.CrossRefGoogle Scholar
9. Harris, T. E. (1963) The Theory of Branching Processes. Springer–Verlag, Berlin.CrossRefGoogle Scholar
10. Heathcote, C. R. (1966) Corrections and comments on the paper ‘A branching process allowing immigration’. J. R. Statist. Soc. B28, 213–217.Google Scholar
11. Hudson, I. L. and Seneta, E. (1977) A note on simple branching processes with infinite mean. J. Appl. Prob. 14, 836–842.CrossRefGoogle Scholar
12. Kawazu, K. and Watanabe, S. (1971) Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 34–51.Google Scholar
13. Knopp, K. (1951) Theory and Applications of Infinite Series. Blackie, London.Google Scholar
14. Pakes, A. G. (1971) Branching processes with immigration. J. Appl. Prob. 8, 32–42.CrossRefGoogle Scholar
15. Pakes, A. G. (1971) On the critical Galton–Watson process with immigration. J. Austral. Math. Soc. 12, 476–482.CrossRefGoogle Scholar
16. Pakes, A. G. (1972) Further results on the critical Galton–Watson process with immigration. J. Austral. Math. Soc. 13, 277–290.Google Scholar
17. Pakes, A. G. (1974) On Markov branching processes with immigration. Sankhyā A37, 129–138.Google Scholar
18. Pakes, A. G. (1975) Some results for non-supercritical Galton–Watson processes with immigration. Math. Biosci. 25, 71–92.CrossRefGoogle Scholar
19. Pakes, A. G. (1975) Some new limit theorems for the critical branching process allowing immigration. Stoch. Proc. Appl. 3, 175–185.Google Scholar
20. Pakes, A. G. (1976) Some limit theorems for a supercritical branching process allowing immigration. J. Appl. Prob. 13, 17–26.CrossRefGoogle Scholar
21. Quine, M. P. (1976) Asymptotic results for estimators in a subcritical branching process with immigration. Ann. Prob. 4, 319–325.CrossRefGoogle Scholar
22. Rényi, A. (1958) On mixing sequences of sets. Acta Math. Acad. Sci. Hungar. 9, 215–228.Google Scholar
23. Rényi, A. and Révész, P. (1958) On mixing sequences of random variables. Acta Math. Acad. Sci. Hungar. 9, 389–393.CrossRefGoogle Scholar
24. Seneta, E. (1969) Functional equations and the Galton–Watson process. Adv. Appl. Prob. 1, 1–42.Google Scholar
25. Seneta, E. (1970) An explicit limit theorem for the critical Galton–Watson process with immigration. J. R. Statist. Soc. B32, 149–152.Google Scholar
26. Seneta, E. (1970) On the supercritical Galton–Watson process with immigration. Math. Biosci. 7, 9–14.CrossRefGoogle Scholar
27. Seneta, E. (1971) On invariant measures for simple branching processes. J. Appl. Prob. 8, 43–51.Google Scholar
28. Seneta, E. (1974) Regularly varying functions in the theory of simple branching processes. Adv. Appl. Prob. 6, 408–420.Google Scholar
29. Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
30. Sevast'yanov, B. A. (1957) Limit theorems for branching processes of special form. Theory Prob. Appl. 2, 321–331.Google Scholar
31. Slack, R. S. (1968) A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.CrossRefGoogle Scholar
32. Titchmarsh, E. C. (1939) The Theory of Functions. Clarendon Press, Oxford.Google Scholar
33. Yang, Y. S. (1972) On branching processes allowing immigration. J. Appl. Prob. 9, 24–31.CrossRefGoogle Scholar