×

Convergence of Markov chain transition probabilities. (English) Zbl 1479.60152

Summary: Consider a discrete time Markov chain with rather general state space which has an invariant probability measure \(\mu \). There are several sufficient conditions in the literature which guarantee convergence of all or \(\mu \)-almost all transition probabilities to \(\mu\) in the total variation (TV) metric: irreducibility plus aperiodicity, equivalence properties of transition probabilities, or coupling properties. In this work, we review and improve some of these criteria in such a way that they become necessary and sufficient for TV convergence of all respectively \(\mu \)-almost all transition probabilities. In addition, we discuss so-called generalized couplings.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G10 Stationary stochastic processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] P. Berti, L. Pratelli, and P. Rigo, Gluing lemmas and Skorohod representations, Electronic Comm. Probab. 20 (2015) 1-11. · Zbl 1330.60010
[2] O. Butkovsky, A. Kulik, and M. Scheutzow, Generalized couplings and ergodic rates for SPDEs and other Markov models, Ann. Appl. Probab. 30 (2020) 1-39. · Zbl 1434.60147
[3] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univ. Press, Cambridge, 1996. · Zbl 0849.60052
[4] R. Douc, E. Moulines, P. Priouret, and P. Soulier, Markov Chains, Springer, Cham, 2018. · Zbl 1429.60002
[5] J. Elstrodt, Maß-und Integrationstheorie, 7th edition, Springer, Berlin, 2011. · Zbl 1259.28001
[6] A. Es-Sarhir, M. v. Renesse, and M. Scheutzow, Harnack inequality for functional SDEs with bounded memory, Electronic Comm. Probab. 14 (2009) 560-565. · Zbl 1195.34124
[7] S. Foss, V. Shneer, J.P. Thomas, and T. Worrall, Stochastic stability of monotone economies in regenerative environments, Journal of Economic Theory 173 (2018) 334-360. · Zbl 1400.91327
[8] M. Hairer and J. Mattingly, Yet another look at Harris’ ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI, Progr. Probab. 63, Birkhäuser, Basel, (2011) 109-117. · Zbl 1248.60082
[9] A. Kulik, Ergodic Behavior of Markov Processes, de Gruyter, Berlin, 2018. · Zbl 1386.60006
[10] A. Kulik and M. Scheutzow, A coupling approach to Doob’s theorem, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015) 83-92. · Zbl 1310.60104
[11] A. Kulik and M. Scheutzow, Generalized couplings and convergence of transition probabilities, Probab. Theory Related Fields 171 (2018) 333-376. · Zbl 1392.60061
[12] S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Second edition, Cambridge Univ. Press, Cambridge, 2009. · Zbl 1165.60001
[13] S. Orey, Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities, Van Nostrand Reinhold, London, 1971. · Zbl 0295.60054
[14] G. O. Roberts and J. S. Rosenthal, General state space Markov chains and MCMC algorithms, Probability Surveys 1 (2004) 20-71. · Zbl 1189.60131
[15] A. Veretennikov, Coupling methods for Markov chains under integral Doeblin type conditions, Theory Stoch. Processes 8 (2002) 383-391
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.