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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.
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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 

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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