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Geometric L2 and L1 convergence are equivalent for reversible Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Gareth O. Roberts  and
Richard L. Tweedie
Gareth O. Roberts*
Affiliation:
University of Lancaster
Richard L. Tweedie*
Affiliation:
University of Minnesota
*
1Postal address: Department of Mathematics and Statistics, University of Lancaster, Lancaster LA1 4YF, England. Email: g.o.roberts@lancaster.ac.uk
†Richard Tweedie died 7 June 2001.
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Abstract

The paper proves the statement of the title, and shows that it has useful applications in evaluating the convergence of queueing models and Gibbs samplers with deterministic and random scans.

Keywords

Rates of convergenceMarkov chainGibbs samplerMCMC

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Type
Markov chains
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 37 - 41
Copyright
Copyright © Applied Probability Trust 2001 

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References

Amit, Y. (1991). On rates of convergence of stochastic relaxation for Gaussian and non-Gaussian distributions. J. Multivar. Anal. 38, 82–99.CrossRefGoogle Scholar
Amit, Y. and Grenander, U. (1991). Comparing sweep strategies for stochastic relaxation. J. Multivar. Anal. 37, 197–222.Google Scholar
Lund, R. L. and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 21, 182–194.Google Scholar
Mengerson, 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
Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for convergence rates of Markov chains. Ann. Appl. Probab. 4, 981–1011.CrossRefGoogle Scholar
Nummelin, E. and Tweedie, R. L. (1978). Geometric ergodicity and R-positivity for general Markov chains. Ann. Probab. 6, 404–420.Google Scholar
Polson, N. G. (1995). Convergence of Markov chain Monte Carlo algorithms. In Bayesian Statistics 5, eds Bernardo, J., Berger, J., Dawid, A. P. and Smith, A. F. M., Oxford University Press, 297–321.Google Scholar
Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Elect. Commun. Probab. 2, 13–25.Google Scholar
Roberts, G. O. and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Proc. Applic. 80, 211–229.Google Scholar
Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90, 558–566.Google Scholar
Vere-Jones, D. (1962). Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford (2) 13, 7–28.Google Scholar
Vere-Jones, D. (1963). On the spectra of some linear operators associated with queueing systems. Z. Wahrscheinlichkeitsth. 2, 12–21.Google Scholar
Vere-Jones, D. (1968). Ergodic properties of non-negative matrices II. Pacific J. Math. 26, 601–620.CrossRefGoogle Scholar
Vere-Jones, D. and Ogata, Y. (1984). On the moments of a self-correcting process. J. Appl. Probab. 21, 335–342.CrossRefGoogle Scholar