Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T21:31:37.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Some properties of a branching process with group immigration and emigration

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.

A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

Keywords

BELLMAN–HARRIS PROCESSTRIBAL EMIGRATIONLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 3 , September 1986 , pp. 628 - 645
Copyright
Copyright © Applied Probability Trust 1986 

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.)

References

Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Brockwell, P. J. (1985) The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Prob. 17, 42–52.Google Scholar
Chover, J., Ney, P. E. and Wainger, S. (1973) Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.Google Scholar
Feller, W. (1969) One-sided analogues of Karamata&s regular variation. L&Enseignement Math. 15, 107–121.Google Scholar
Goldstein, M. I. and Hoppe, F. M. (1981) Necessary and sufficient lifetime conditions for normed convergence of critical age-dependent processes with infinite variance. Ann. Prob. 9, 490–497.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, New York.Google Scholar
Kaplan, N. (1975) Limit theorems for a GI/G/8 queue. Ann. Prob. 3, 780–789.CrossRefGoogle Scholar
Kaplan, N. and Pakes, A. G. (1974) Super critical age-dependent branching processes with immigration. Stoch. Proc. Appl. 2, 371–390.Google Scholar
Pakes, A. G. (1975a) On Markov branching processes with immigration. Sankhya A37, 129–138.Google Scholar
Pakes, A. G. (1975b) Limit theorems for the integrals of some branching processes. Stoch. Proc. Appl. 3, 89–111.Google Scholar
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
Pakes, A. G. and Kaplan, N. (1974) On the subcritical Bellman-Harris process with immigration. J. Appl. Prob. 11, 652–668.Google Scholar
Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
Slack, R. S. (1968) A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.Google Scholar
Tessera, A. (1984) Population processes allowing emigration of families. J. Appl. Prob. 21, 225–232.Google Scholar
Yang, Y. S. (1972) On branching processes allowing immigration. J. Appl. Prob. 9, 24–31.Google Scholar