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Comments on the age distribution of Markov processes

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes  and
Simon Tavaré
Anthony G. Pakes*
Affiliation:
University of Western Australia
Simon Tavaré
Affiliation:
University of Utah
*
∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA6009, Australia.
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Abstract

Previous work on the concept of a limiting conditional age distribution of a discrete-state continuous-time Markov process with one absorbing state is generalised. The generalisation allows this process to have a finite number of absorbing states and the associated return process to have an arbitrary initial distribution on the transient states of the absorbing process. If the return process is ρ-recurrent, possesses the strong ratio limit property and satisfies some further requirements then the limiting age distribution exists. The proof of this result requires a new representation of the ρ-invariant measure of the return process.

The following examples are treated, (a) finite state space birth-death processes, (b) Markov branching processes and the linear death process, and (c) the linear birth and death process with killing.

Keywords

MARKOV PROCESSLIMITING AGEρ-CLASSIFICATIONSTRONG RATIO LIMIT PROPERTYBIRTH AND DEATH PROCESSESMARKOV BRANCHING PROCESSESLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 681 - 703
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

a

Present address: Department of Statistics, Colorado State University, Fort Collins, Colorado, 80523, U.S.A.

References

Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, Berlin.Google Scholar
David, H. A. (1970) Order Statistics, Wiley, New York.Google Scholar
Erickson, K. B. (1970) Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151, 263–291.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Garsia, A. M. (1963) Some Tauberian theorems and the asymptotic behaviour of probabilities of recurrent events. J. Math. Anal. Appl. 7, 146–162.Google Scholar
Hall, P. (1979) On the relative stability of large order statistics. Math. Proc. Camb. Phil. Soc. 86, 467–475.Google Scholar
Karlin, S. and McGregor, J. (1962) On a genetics model of Moran. Prob. Camb. Phil. Soc. 58, 299–311.CrossRefGoogle Scholar
Karlin, S. and Tavaré, S. (1980) Non-random mating and its effect on the time to production at particular genotypes in finite populations. In preparation.Google Scholar
Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton-Watson process with mean one and finite variance. Theory Prob. Appl. 11, 513–540.Google Scholar
Kimura, M. and Ohta, T. (1973) The age of a neutral mutant persisting in a finite population. Genetics 75, 199–212.Google Scholar
Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. Lond. Math. Soc. (3) 13, 337–358.Google Scholar
Levikson, B. (1977) The age distribution of Markov processes. J. Appl. Prob. 14, 492–506.Google Scholar
Nagasawa, M. and Maruyama, T. (1979) An application of time reversal of Markov processes to a problem of population genetics. Adv. Appl. Prob. 11, 457–478.Google Scholar
Nummelin, E. (1979) Strong ratio limit theorems for φ-recurrent Markov chains. Ann. Prob. 7, 639–650.Google Scholar
Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Probabilities. Van Nostrand-Reinhold Mathematical Studies No. 34, London.Google Scholar
Pakes, A. G. (1971) A branching process with a state dependent immigration component. Adv. Appl. Prob. 3, 301–314.Google Scholar
Pakes, A. G. (1975) On Markov branching processes with immigration. Sankhyā A 37, 129–138.Google Scholar
Pakes, A. G. (1979) The age of a Markov process. Stoch. Proc. Appl. 8, 277–303.Google Scholar
Stewart, D. B. (1976) Branching Processes with State-dependent Immigration. , Monash University.Google Scholar
Tavaré, S. (1979) Some Results for Markov Processes with Application to Genetic Models. , University of Sheffield.Google Scholar
Tavaré, S. (1980) Time reversal and age distributions I. Discrete Markov chains. J. Appl. Prob. 17, 33–46.Google Scholar
Tweedie, R. L. (1974) Some ergodic properties of the Feller minimal process. Quart. J. Math. Oxford. 25, 485–495.Google Scholar
Watterson, G. A. (1976) Reversibility and the age of an allele. I. Moran's infinitely many neutral alleles model. Theoret. Popn Biol. 10, 139–253.Google Scholar
Watterson, G. A. (1977) Reversibility and the age of an allele II. Two-allele models, with selection and mutation. Theoret. Popn Biol. 12, 179–196.Google Scholar
Yamazato, M. (1975) Some results on continuous time branching process with state-dependent immigration. J. Math. Soc. Japan. 17, 479–496.Google Scholar