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Moments for stationary and quasi-stationary distributions of markov chains

Published online by Cambridge University Press:  14 July 2016

E. Seneta  and
R. L. Tweedie
E. Seneta*
Affiliation:
University of Sydney
R. L. Tweedie*
Affiliation:
SIROMATH Pty Ltd
*
∗Postal address: Department of Mathematical Statistics, University of Sydney, Sydney, NSW 2006, Australia.
∗∗Postal address: SIROMATH Pty Ltd, 31 Market St, Sydney, NSW 2000, Australia.
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Abstract

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

Keywords

IRREDUCIBLE R-POSITIVE MATRIXR-INVARIANT MEASURETABOO STATESMOMENTSSUBCRITICAL BRANCHING PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 148 - 155
Copyright
Copyright © Applied Probability Trust 1985 

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References

[1] Bagley, J. H. (1982) Asymptotic properties of subcritical Galton–Watson processes. J. Appl. Prob. 19, 510–517.Google Scholar
[2] Buiculescu, M. (1975) On quasi-stationary distributions for multi-type Galton–Watson processes. J. Appl. Prob. 12, 60–68.Google Scholar
[3] Nummelin, E. and Tweedie, R. L. (1978) Geometric ergodicity and R -positivity for general Markov chains. Ann. Prob. 6, 404–420.Google Scholar
[4] Seneta, E. (1981) Non-Negative Matrices and Markov Chains. Springer-Verlag, New York.CrossRefGoogle Scholar
[5] Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.CrossRefGoogle Scholar
[6] Tweedie, R. L. (1974) R-theory for Markov chains on general state space I: Solidarity properties and R -recurrent chains. Ann. Prob. 2, 840–864.Google Scholar
[7] Tweedie, R. L. (1974) Quasi-stationary distributions for Markov chains on a general state space. J. Appl. Prob. 11, 726–741.CrossRefGoogle Scholar
[8] Tweedie, R. L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.Google Scholar