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Taboo extinction, sojourn times, and asymptotic growth for the Markovian birth and death process

Published online by Cambridge University Press:  20 February 2017

W. A. O'n. Waugh
W. A. O'n. Waugh*
Affiliation:
University of Toronto
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Abstract

A well-known result in the theory of branching processes provides an asymptotic expression for the population size (valid for large times) in terms of a single random variable, multiplied by a deterministic exponential growth factor. In the present paper this is generalized to a class of size-dependent population models. The work is based on the series of sojourn times. An essential tool is the use of probabilities conditional upon non-extinction (taboo probabilities).

Keywords

MARKOV BIRTH AND DEATH PROCESSTABOO STATESSOJOURN TIMESBRANCHING PROCESSESASYMPTOTIC POPULATION GROWTHEXTINCTION OF POPULATIONSTOCHASTIC LAGEXPLOSIVE MARKOV PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 3 , June 1972 , pp. 486 - 506
Copyright
Copyright © Applied Probability Trust 1972 

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References

Breny, H. (1962) Cheminements conditionnels de chaînes de Markov absorbantes. Ann. Soc. Sci. Bruxelles 76, 81–87.Google Scholar
Dobrušin, R. L. (1952) On conditions of regularity of stationary Markov processes with a denumerable number of possible states. Uspehi Mat. Nauk. (N.S.) 7 no. 6, 185–191. (In Russian).Google Scholar
Feller, W. (1968), (1970) An Introduction to Probability Theory and its Applications. Vol. 1, 3rd edition, Vol. 2, 2nd edition, John Wiley and Sons, Inc., NewYork.Google Scholar
Hoem, J. M. (1969) Purged and partial Markov chains. Skand. Aktuarietidskr. 52, 147–155.Google Scholar
Jensen, A. (1954) A Distribution Model Applicable to Economics. Munksgaard, Copenhagen.Google Scholar
John, P. W. M. (1957) Divergent time homogeneous birth and death processes. Ann. Math. Statist. 28, 514–517.Google Scholar
Karlin, S. and Mcgregor, J. (1955) Representation of a class of stochastic processes. Proc. Nat. Acad. Sci. 41, 387–391.Google Scholar
Kato, T. (1954) On the semigroups generated by Kolmogorov's differential equations. J. Math. Soc. Japan 6, 1–15.Google Scholar
Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton.Google Scholar
Kendall, D. G. and Reuter, G. E. H. (1957) The calculation of the ergodic projection for Markov chains and processes with a countable infinity of states. Acta Math. 97, 103–144.Google Scholar
Reuter, G. E. H. and Ledermann, W. (1953) On the differential equations for the transition probabilities of Markov processes with enumerably many states. Proc. Camb. Phil. Soc. 49, 247–262.Google Scholar
Waugh, W. A. O'N. (1958) Conditioned Markov processes. Biometrika 45, 241–249.CrossRefGoogle Scholar
Waugh, W. A. O'N. (1970) Uses of the sojourn time series for the Markovian birth process. Proc. Sixth Berkeley Symposium on Math. Statist. and Prob. Google Scholar