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

  • F. C. Klebaner (a1)

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.

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Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, VIC 3052, Australia.

References

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[1] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.
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[4] Loève, M. (1955) Probability Theory. Foundations. Random Sequences. Van Nostrand, New York.
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[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, 2938.
[7] Teicher, H. (1980) Almost certain behaviour of row sums of double arrays. Conf. Analytic Methods for Probability Theory, Oberwolfach, Springer-Verlag, Berlin.

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