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A note on the geometric ergodicity of a Markov chain

Published online by Cambridge University Press:  01 July 2016

K. S. Chan
K. S. Chan*
Affiliation:
University of Chicago
*
∗Postal address: Department of Statistics, The University of Chicago, Chicago, IL 60637, USA.
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Abstract

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It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b> 0 and ρ < 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.

Keywords

AUTOREGRESSIVE PROCESSDRIFT CONDITIONRANDOM WALK
Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 702 - 704
Copyright
Copyright © Applied Probability Trust 1989 

References

Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press.Google Scholar
Nummelin, E. and Tuominen, P. (1982) Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Proc. Appl. 12, 187–202.Google Scholar
Tweedie, R. L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar