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Locally contracting iterated functions and stability of Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

S. F. Jarner  and
R. L. Tweedie
S. F. Jarner*
Affiliation:
Lancaster University
R. L. Tweedie*
Affiliation:
University of Minnesota
*
∗ Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK.
∗∗ Postal address: Division of Biostatistics, School of Public Health, A460 Mayo Building, Box 303 420 Delaware Street, SE Minneapolis, MN 55455-0378, USA. Email address: tweedie@biostat.umn.edu
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Abstract

We consider Markov chains in the context of iterated random functions and show the existence and uniqueness of an invariant distribution under a local contraction condition combined with a drift condition, extending results of Diaconis and Freedman. From these we deduce various other topological stability properties of the chains. Our conditions are typically satisfied by, for example, queueing and storage models where the global Lipschitz condition used by Diaconis and Freedman normally fails.

Keywords

Markov chainsiterated functionsgeometric convergencestochastic monotonicityrates of convergence

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 494 - 507
Copyright
Copyright © Applied Probability Trust 2001 

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Footnotes

Work supported in part by NSF grant DMS 9803682 and EPSRC grant GR/J19900.

References

Borovkov, A. A., and Foss, S. G. (1992). Stochastically recursive sequences and their generalizations. Siberian Adv. Math. 2, 16–81.Google Scholar
Diaconis, P., and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 45–76.Google Scholar
Jarner, S. F., and Tweedie, R. L. (2001). Stability properties of Markov chains defined via iterated random functions. Submitted.Google Scholar
Kifer, Ju. (1986). Ergodic Theory of Random Transformations. Birkhäuser, Basel.Google Scholar
Lund, R. B., and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 20, 182–194.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1993). The Doeblin decomposition. In Doeblin and Modern Probability (Contemp. Math. 149), ed. Cohn, H. American Mathematical Society, Providence, RI, pp. 211–225.CrossRefGoogle Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.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.CrossRefGoogle Scholar
Roberts, G. O., and Rosenthal, J. S. (1997). Shift-coupling and convergence rates of ergodic averages. Stoch. Models 13, 147–165.Google Scholar
Roberts, G. O., and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Proc. Appl. 80, 211–229.Google Scholar
Roberts, G. O., and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone stochastic processes. J. Appl. Prob. 37, 359–373.Google Scholar
Steinsaltz, D. (1999). Locally contractive iterated function systems. Ann. Prob. 27, 1952–1979.CrossRefGoogle Scholar
Wu, W. B. (2000). Iterated function systems. Part I. Stationarity and path properties. Tech. Rept, University of Michigan.Google Scholar
Wu, W. B., and Woodroofe, M. (1999). A central limit theorem for iterated random functions. Tech. Rept, University of Michigan.Google Scholar