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Population-size-dependent branching process with linear rate of growth

Published online by Cambridge University Press:  14 July 2016

F. C. Klebaner
F. C. Klebaner*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, VIC 3052, Australia.
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Abstract

The process we consider is a binary splitting, where the probability of division, , depends on the population size, 2i. We show that Zn converges to ∞ almost surely on a set of positive probability. Zn /n converges in distribution to a proper limit, diverges almost surely on converges almost surely on and there are no constants cn such that Zn /cn converges in probability to a non-degenerate limit.

Keywords

MARKOV CHAINSPOPULATION MODELSTATE DEPENDENCEGROWTH RATES

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 242 - 250
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[2] Labkovskii, V. A. (1972) A limit theorem for generalized random branching process depending on the size of the population. Theory Prob. Appl. 17, 72–85.CrossRefGoogle Scholar
[3] Levina, L. V., Leontovich, A. M. and Piatetski-Shapiro, I. I. (1968) On a regulative branching process. Problemy Pederaci Informatsii 4, 72–82.Google Scholar
[4] Loève, M. (1955) Probability Theory. Foundations. Random Sequences. Van Nostrand, New York.Google Scholar
[5] Rao, K. S. and Kendall, D. G. (1950) On the generalized second limit-theorem in the calculus of probabilities. Biometrika 37, 224–230.CrossRefGoogle ScholarPubMed
[6] Schuh, H.-J. (1982) Sums of i.i.d. random variables and an application to the explosion criterion for Markov branching processes. J. Appl. Prob. 19, 29–38.CrossRefGoogle Scholar
[7] Teicher, H. (1980) Almost certain behaviour of row sums of double arrays. Conf. Analytic Methods for Probability Theory, Oberwolfach, Springer-Verlag, Berlin.Google Scholar