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Convergence of Markov chains in the relative supremum norm

Part of: Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Lars Holden
Lars Holden*
Affiliation:
Norwegian Computing Center
*
∗ Postal address: Norwegian Computing Center, P.O. Box 114, Blindern, N-0314 Oslo, Norway. Email address: lars.holden@nr.no
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Abstract

It is proved that the strong Doeblin condition (i.e., ps(x,y) ≥ asπ(y) for all x,y in the state space) implies convergence in the relative supremum norm for a general Markov chain. The convergence is geometric with rate (1 - as)1/s. If the detailed balance condition and a weak continuity condition are satisfied, then the strong Doeblin condition is equivalent to convergence in the relative supremum norm. Convergence in other norms under weaker assumptions is proved. The results give qualitative understanding of the convergence.

Keywords

Markov chaingeometric convergencerelative supremum norm

MSC classification

Primary: 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1074 - 1083
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

Research supported by the Research Council of Norway.

References

Cressie, N. A. C. (1991). Statistics for Spatial Data. John Wiley, New York.Google Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Geyer, C. J. and Möller, J. (1993). Simulation procedures and likelihood inference for spatial point processes. Tech. Rept 260, Dept Theoret. Statist., Inst. Math., University of Aarhus.Google Scholar
Gilks, W. R., Richardson, S., and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall, London.Google Scholar
Holden, L. (1998). Geometric convergence of the Metropolis–Hastings simulation algorithm. Statist. Prob. Lett. 39, 371–377.CrossRefGoogle Scholar
Mengersen, K. L., and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24, 101–121.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Ripley, B. D. (1987). Stochastic Simulation. John Wiley, New York.Google Scholar