Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-04T12:24:34.932Z Has data issue: false hasContentIssue false
  • English
  • Français

Limiting distributions for the general branching process with immigration

Published online by Cambridge University Press:  14 July 2016

S. D. Durham
S. D. Durham*
Affiliation:
University of South Carolina
Get access Rights & Permissions [Opens in a new window]

Abstract

For the general age-dependent branching process subject to an arbitrary pattern of immigration asymptotic distributions are studied in the supercritical and critical cases. In the supercritical case a new functional form is given for the limiting cumulant-generating function and in the critical case moment conditions are given under which a randomized gamma limit law prevails. The age-structure is also considered.

Keywords

AGE-DEPENDENT BRANCHING PROCESSESIMMIGRATIONFUNCTIONAL EQUATIONSLIMITING DISTRIBUTIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 809 - 813
Copyright
Copyright © Applied Probability Trust 1974 

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

Crump, K. and Mode, C. J. (1969) A general age-dependent branching process II. J. Math. Anal. Appl. 25, 8–17.Google Scholar
Doney, R. A. (1972) A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.Google Scholar
Doney, R. A. (1973) On a functional equation for general branching processes. J. Appl. Prob. 10, 198–205.Google Scholar
Durham, S. D. (1971) Limit theorems for a general critical branching process. J. Appl. Prob. 8, 1–16.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
Foster, J. (1972) The asymptotic behavior of the critical age-dependent branching process with immigration. Preprint.Google Scholar
Ganuza, E. Z. (1973) On the supercritical branching process. J. Appl. Prob. Submitted.Google Scholar
Ganuza, E. Z. and Foster, J. (1973) Private communication.Google Scholar
Jagers, P. (1969) A general stochastic model for population development. Skand. Aktuartidskr. 52, 84–103.Google Scholar
Pakes, A. G. (1972) Limit theorems for age-dependent branching processes with immigration. Math. Biosci. 14, 221–234.Google Scholar
Radcliffe, J. (1972a) The convergence of a generalized multiple age-dependent branching process with Poisson immigration. Math. Biosci. 13, 125–132.Google Scholar
Radcliffe, J. (1972b) The convergence of supercritical age-dependent branching processes allowing immigration at the epochs of a renewal process. Math. Biosci. 14, 37–44.Google Scholar
Weiner, H. J. (1972a) A critical age-dependent branching process with immigration. Ann. Math. Statist. 43, 2099–2103.Google Scholar
Weiner, H. J. (1972b) A multitype critical age-dependent branching process with immigration. J. Appl. Prob. 9, 697–706.Google Scholar