Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T05:32:36.492Z Has data issue: false hasContentIssue false
  • English
  • Français

On recurrence and transience of growth models

Published online by Cambridge University Press:  14 July 2016

G. Kersting
G. Kersting*
Affiliation:
J. W. Goethe-Universität, Frankfurt
*
∗Postal address: Fachbereich Mathematik, J. W. Goethe-Universität, D-6000 Frankfurt, W. Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Xn be non-negative random variables, possessing the Markov property. We given criteria for deciding whether Pr(Xn →∞) is positive or 0. It turns out that essentially this depends on the magnitude of E(Xn+1 | Xn = x) compared to that of E(X2n+1 | Xn = x) for large x. The assumptions are chosen such that for example population-dependent branching processes can be treated by our results.

Keywords

LYAPUNOV FUNCTIONMARTINGALESMARKOV PROPERTYSTATE-DEPENDENT BRANCHING PROCESS
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 614 - 625
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.)

Footnotes

Research partially supported by the SFB 123, ‘Stochastische Mathematische Modelle', Heidelberg.

References

[1] Hall, P. and Heyde, C. C. (1980) Martingale Theory and its Applications. Academic Press, New York.Google Scholar
[2] Keller, G., Kersting, G. and Rösler, U. (1985) On the asymptotic behavior of time-discrete stochastic growth processes. Preprint.Google Scholar
[3] Klebaner, F. C. (1984) On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.Google Scholar
[4] Lamperti, J. (1960) Criteria for recurrence or transience of stochastic processes. J. Math. Anal. Appl. 1, 314–330.CrossRefGoogle Scholar
[5] Levy, J. B. (1979) Transience and recurrence of state-dependent branching processes with an immigration component. Adv. Appl. Prob. 11, 73–92.CrossRefGoogle Scholar
[6] Zubkov, A. M. (1974) Analogies between Galton–Watson processes and f -branching processes. Theory Prob. Appl. 19, 309–331.Google Scholar