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The existence of moments for stationary Markov chains

Published online by Cambridge University Press:  14 July 2016

R. L. Tweedie
R. L. Tweedie*
Affiliation:
Siromath, Sydney
*
∗ Postal address: SIROMATH Pty Ltd, 71 York Street, Sydney, NSW 2000, Australia.
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Abstract

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx . In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

Keywords

INVARIANT MEASUREGEOMETRIC ERGODICITYMOMENT-GENERATING FUNCTIONHARRIS CHAIN

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 191 - 196
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Nummelin, E. (1978) A splitting technique for Harris recurrent Markov chains. Z. Wahrscheinlichkeitsth. 43, 309–318.CrossRefGoogle Scholar
[2] Nummelin, E. and Tuominen, P. (1982) Geometric ergodicity of Harris recurrent Markov chains with application to renewal theory. Stoch. Proc. Appl. 12, 187–202.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] Pollard, D. B. and Tweedie, R. L. (1976) R-theory for Markov chains on a topological state space II. Z. Wahrscheinlichkeitsth. 34, 269–278.Google Scholar
[5] Tuominen, P. and Tweedie, R. L. (1979) Markov chains with continuous components. Proc. London Math. Soc. (3) 38, 89–114.CrossRefGoogle Scholar
[6] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar
[7] Tweedie, R. L. (1982) Criteria for rates of convergence of Markov chains, with application to queueing theory. In Papers in Probability, Statistics and Analysis, ed. Kingman, J. F. C. and Reuter, G. E. H., Cambridge University Press, London.Google Scholar