Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-22T05:30:39.048Z Has data issue: false hasContentIssue false
  • English
  • Français

Some limit theorems for continuous-state branching processes

Part of: Markov processes

Published online by Cambridge University Press:  09 April 2009

Anthony G Pakes
Anthony G Pakes
Affiliation:
Department of Mathematics University of Western Australia, Nedlands, 6009Western AustraliaAustralia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The most general continuous time and state branching (C.B.) process (Xt) can be constructed as a certain random time transformation of a spectrally positive Levy process. When the generating process is compound Poisson with a superimposed negative linear drift and the C.B. process is not supercritical, then there is a random time T such that Xt+T = e-ctXT where c > 0 is the drift parameter. Thus T is the last epoch of random variation.

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 1 , February 1988 , pp. 71 - 87
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Athreya, K. B. and Ney, P. E., Branching processes (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
[2]Breiman, L., Probability (Addision-Wesley, Reading, Mass., 1968).Google Scholar
[3]Grey, D. R., ‘Asymptotic behavior of continuous time, continuous state-space branching processes’, J. Appl. Probab. 11 (1974), 669–677.CrossRefGoogle Scholar
[4]Kallenberg, P. J. M., Branching processes with continuous state space (Math. Centrum, Amsterdam, 1979).Google Scholar
[5]Kingman, J. F. C., ‘Ergodic properties of continuous-time Markov processes and their discrete skeletons’, Proc. London Math. Soc. 13 (1963), 593–604.CrossRefGoogle Scholar
[6]Kuczma, M., Functional equations in a single variable (PWN, Warsaw, 1968).Google Scholar
[7]Pakes, A. G., ‘Some limit theorems for Jifina process’, Period. Math. Hungar. 10 (1979), 55–66.CrossRefGoogle Scholar
[8]Pakes, A. G. and Trajstman, A. C., ‘Some properties of continuous-state branching processes, with applications to Bartoszynski's virus model’, Adv. in Appl. Probab. 17 (1985), 23–41.CrossRefGoogle Scholar
[9]Seneta, E., ‘The Galton-Watson process with mean one’, J. Appl. Probab. 4 (1967), 489–495.CrossRefGoogle Scholar
[10]Seneta, E., ‘Some supplementary notes on one-type continuous-state branching processes’, Z. Wahrsch. Verw. Getoete 34 (1976), 87–89.CrossRefGoogle Scholar
[11]Seneta, E. and Vere-Jones, D., ‘On the asymptotic behavior of subcritical branch ing processes with continuous state space’, Z. Wahrsch. Verw. Gebiete 10 (1968), 212–225.CrossRefGoogle Scholar
[12]Seneta, E. and Vere-Jones, D., ‘On a problem of M. Jirina concerning continuous state branching processes’, Czech. Math. J. 19(94) (1969), 277–283.CrossRefGoogle Scholar