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The asymptotic behaviour of birth and death and some related processes

Published online by Cambridge University Press:  01 July 2016

Andrew D. Barbour
Andrew D. Barbour*
Affiliation:
University of Cambridge
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Abstract

The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.

Keywords

BIRTH AND DEATHCENTRAL LIMITITERATED LOGARITHM
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 1 , March 1975 , pp. 28 - 43
Copyright
Copyright © Applied Probability Trust 1975 

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References

Barbour, A. D. (1973) Limit theorems for Markov population processes. , University of Cambridge.Google Scholar
Barbour, A.D. (1974) Tail sums of convergent series of independent random variables. Proc. Camb. Phil. Soc. 75, to apper.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Chover, J. (1967) On Strassen's version of the loglog law. Z. Wahrscheinlichkeitsth. 8, 83–90.CrossRefGoogle Scholar
Chung, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.Google Scholar
Karlin, S. and McGregor, J. L. (1957) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.CrossRefGoogle Scholar
Kendall, D. G. (1966) Branching processes since 1873. J. Lond. Math. Soc. 41, 385–406.CrossRefGoogle Scholar
Loève, M. (1955) Probability Theory. Van Nostrand, New York.Google Scholar
Stout, W. F. (1970) A martingale analogue of Kolmogorov's law of the iterated logarithm. Z. Wahrscheinlichkeitsth. 15, 279–290.CrossRefGoogle Scholar
Waugh, W. A. O'N. (1970) Transformations of a birth process into a Poisson process. J. R. Statist. Soc. B 32, 418–431.Google Scholar
Waugh, W. A. O'N. (1972) Taboo extinction, sojourn times and asymptotic growth for the Markovian birth and death process. J. Appl. Prob. 9, 486–506.CrossRefGoogle Scholar