Abstract
In this paper, we obtain the quantitative bound of the exponential convergence rates of Markov chains under a weaken minorization condition, using the coupling method and the analytic approach. And also, we obtain the convergence rates for continuous time Markov processes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, M. F.: Eigenvalues, Inequalities, and Ergodic Theory, Springer, London, 2005
Douc, R., Fort, G., Guillin, A.: Subgeometric rates of convergence of f-ergodic strong Markov processes. Stochastic Process. Appl., 119(3), 897–923 (2009)
Douc, R., Moulines, E., Rosenthal, J. S.: Quantitative bounds on convergence of time-inhomogeneous Markov chains. Stochastic Process. Appl., 14(4), 1643–1665 (2004)
Hairer, M., Mattingly, J. C.: Yet another look at Harris’ ergodic theorem for Markov chains. Seminar on Stochastic Analysis, Random Fields and Applications VI, 109C117, Progr. Probab., 63, 2011
Hairer, M., Mattingly, J. C., Scheutzow, M.: Asymptotic coupling and a general form of Harris theorem with applications to stochastic delay equations. Probability Theory and Related Fields, 149(1-2), 223–259 (2009)
Ledoux, M.: The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001
Meyn, S. P., Tweedie, R. L.: Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, 2009
Moulines, E., Douc, R., Soulier, P.: Computable convergence rates for sub-geometric ergodic Markov chains. Bernoulli, 13(3), 831–848 (2007)
Roberts, G. O., Rosenthal, J. S.: Quantitative bounds for convergence rates of continuous time Markov processees. EJP, 1(9), 1–21 (1996)
Rosenthal, J. S.: Minorization conditions and convergence rates for Markov chain monte carlo. Journal of the American Statistical Association, 90(430), 558–566 (1995)
Rosenthal, J. S.: Quantitative convergence rates of Markov chains: A simple account. Electronic Communications in Probability, 7, 123–128 (2002)
Wang, F. Y.: Functional Inequalities, Markov Processes and Spectral Theory, Science Press, Beijing, 2005
Acknowledgements
The authors would like to thank Prof. Liming WU for lots of helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (Grant No. 11771161) and the National Social Science Fund of China (Grant No. 17BTJ034)
Rights and permissions
About this article
Cite this article
Hu, S.L., Wang, X.Y. Exponential Convergence Rates of Markov Chains under a Weaken Minorization Condition. Acta. Math. Sin.-English Ser. 34, 1829–1836 (2018). https://doi.org/10.1007/s10114-018-7541-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-018-7541-8