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On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states

Published online by Cambridge University Press:  14 July 2016

E. Seneta  and
D. Vere-Jones
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Abstract

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.

Type
Research Papers
Information
Journal of Applied Probability , Volume 3 , Issue 2 , December 1966 , pp. 403 - 434
Copyright
Copyright © Applied Probability Trust 

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References

[1] Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discrete finite Markov chains. J. Appl. Prob. 2, 88–100.Google Scholar
[2] Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. 1 John Wiley, New York.Google Scholar
[3] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
[4] Heathcote, C. R., Seneta, E. and Vere-Jones, D. A refinement of two theorems in the theory of branching processes (submitted for publication).Google Scholar
[5] Karlin, S. and Mcgregor, J. L. (1959) Random walks. Illinois J. Math. 3, 66–81.CrossRefGoogle Scholar
[6] Kendall, D. G. (1966) On super-critical branching processes with a positive chance of extinction. To appear in Festschrift Volume in honour of J. Neyman.Google Scholar
[7] Kendall, D. G. (1966) Proposal of thanks in the discussion of [11] (below). To appear in J. Roy. Statist. Soc. Ser. B 28, 266–269.Google Scholar
[8] Kolmogorov, A. N. (1938) Zur Lösung einer biologischen Aufgabe. Comm. Math. Mech. Chebychev Univ., Tomsk 2, 1–6.Google Scholar
[9] Mandl, P. (1959) Sur le comportement asymptotique des probabilités dans les ensembles des états d'une chaîne de Markov homogène (in Russian). Casopis Pest. Mat. 84, 140–149.Google Scholar
[10] Pruitt, W. E. (1964) Eigenvalues of non-negative matrices. Ann. Math. Statist. 35, 1797–1800.Google Scholar
[11] Seneta, E. (1966) Quasi-stationary distributions and time-reversion in genetics (with discussion). J. Roy. Statist. Soc. Ser. B 28, 253–277.Google Scholar
[12] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. (2) 13, 7–28.Google Scholar
[13] Vere-Jones, D. Ergodic properties of non-negative matrices (submitted for publication).Google Scholar
[14] Yaglom, A. M. (1947) Certain limit theorems of the theory of branching stochastic processes (in Russian). Dokl. Akad. Nauk SSSR (n.s.) 56, 795–798.Google Scholar