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Quantitative Convergence Rates for Subgeometric Markov Chains

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Christophe Andrieu ,
Gersende Fort  and
Matti Vihola
Christophe Andrieu*
Affiliation:
University of Bristol
Gersende Fort*
Affiliation:
CNRS and Télécom ParisTech
Matti Vihola*
Affiliation:
University of Jyväskylä
*
∗ Postal address: School of Mathematics, University of Bristol, Bristol BS8 1TW, UK.
∗∗ Postal address: Télécom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France.
∗∗∗ Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, Jyväskylä, FI-40014, Finland Email address: matti.s.vihola@jyu.fi
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Abstract

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We provide explicit expressions for the constants involved in the characterisation of ergodicity of subgeometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The results are fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our results accommodate also some classes of inhomogeneous chains.

Keywords

Markov chaininhomogeneoussubgeometric ergodicitypolynomial ergodicity

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J22: Computational methods in Markov chains
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 391 - 404
Copyright
© Applied Probability Trust 

References

Andrieu, C. and Fort, G. (2005). “Explicit control of subgeometric ergodicity.” Res. Rep. 05:17, School of Mathematics, University of Bristol.Google Scholar
Andrieu, C. and Roberts, G. O. (2009). “The pseudo-marginal approach for efficient Monte Carlo computations.” Ann. Statist. 37, 697–725.CrossRefGoogle Scholar
Andrieu, C. and Thoms, J. (2008). “A tutorial on adaptive MCMC.” Statist. Comput. 18, 343–373.CrossRefGoogle Scholar
Andrieu, C. and Vihola, M. (2014). “Markovian stochastic approximation with expanding projections.” Bernoulli 20, 545–585.CrossRefGoogle Scholar
Andrieu, C. and Vihola, M. (2015). “Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms.” Ann. Appl. Prob. 25, 1030–1077.CrossRefGoogle Scholar
Andrieu, C., Moulines, É. and Priouret, P. (2005). “Stability of stochastic approximation under verifiable conditions.” {SIAM J. Control Optimization} 44, 283–312.CrossRefGoogle Scholar
Atchadé, Y. and Fort, G. (2010). “Limit theorems for some adaptive MCMC algorithms with subgeometric kernels.” Bernoulli 16, 116–154.CrossRefGoogle Scholar
Baxendale, P. H. (2005). “Renewal theory and computable convergence rates for geometrically ergodic Markov chains.” Ann. Appl. Prob. 15, 700–738.CrossRefGoogle Scholar
Douc, R., Moulines, E. and Rosenthal, J. S. (2004). “Quantitative bounds on convergence of time-inhomogeneous Markov chains.” Ann. Appl. Prob. 14, 1643–1665.CrossRefGoogle Scholar
Douc, R., Moulines, E. and Soulier, P. (2007). “Computable convergence rates for sub-geometric ergodic Markov chains.” Bernoulli 13, 831–848.CrossRefGoogle Scholar
Douc, R., Fort, G., Moulines, E. and Soulier, P. (2004). “Practical drift conditions for subgeometric rates of convergence.” Ann. Appl. Prob. 14, 1353–1377.CrossRefGoogle Scholar
Fort, G. (2001). “Contrôle explicite d'ergodicité de chaîne de Markov: applications à l'analyse de convergence de l'algorithme Monte-Carlo EM.” , Université Paris VI.Google Scholar
Fort, G. and Moulines, E. (2003). “Polynomial ergodicity of Markov transition kernels.” Stoch. Process. Appl. 103, 57–99.CrossRefGoogle Scholar
Fort, G., Moulines, E. and Priouret, P. (2011). “Convergence of adaptive and interacting Markov chain Monte Carlo algorithms.” Ann. Statist. 39, 3262–3289.CrossRefGoogle Scholar
Jarner, S. F. and Roberts, G. O. (2002). “Polynomial convergence rates of Markov chains.” Ann. Appl. Prob. 12, 224–247.CrossRefGoogle Scholar
Maire, F., Douc, R. and Olsson, J. (2014). “Comparison of asymptotic variances of inhomogeneous Markov Chains with application to Markov chain Monte Carlo methods.” Ann. Statist. 42, 1484–1510.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1994). “Computable bounds for geometric convergence rates of Markov chains.” Ann. Appl. Prob. 4, 981–1011.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R. L. (2009). “Markov Chains and Stochastic Stability, 2nd edn.” Cambridge University Press.CrossRefGoogle Scholar
Saksman, E. and Vihola, M. (2010). “On the ergodicity of the adaptive Metropolis algorithm on unbounded domains.” Ann. Appl. Prob. 20, 2178–2203.CrossRefGoogle Scholar
Tuominen, P. and Tweedie, R. L. (1994). “Subgeometric rates of convergence of f-ergodic Markov chains.” Adv. Appl. Prob. 26, 775–798.CrossRefGoogle Scholar