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On some basic features of strictly stationary, reversible Markov chains

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  • Richard C. Bradley

Abstract

It has been well known for some time that for strictly stationary Markov chains that are ‘reversible’, the special symmetry (with the distribution of the Markov chain as a whole being invariant under a reversal of the ‘direction of time’) provides special extra features in the mathematical theory. This article here is in part an exposition of some of the basic aspects of that special theory. The mathematical techniques employed in this review are relatively gentle, involving only some basic measure‐theoretic probability theory. To that special theory, a couple of new results are contributed here that are connected with the Rosenblatt strong mixing condition; and those new results in turn assist in bringing further clarity to the exposition of the theory.

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  • Richard C. Bradley, 2021. "On some basic features of strictly stationary, reversible Markov chains," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 499-533, September.
    • Handle: RePEc:bla:jtsera:v:42:y:2021:i:5-6:p:499-533
    DOI: 10.1111/jtsa.12583
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    References listed on IDEAS

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    1. Longla, Martial & Peligrad, Magda, 2012. "Some aspects of modeling dependence in copula-based Markov chains," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 234-240.
    2. Brendan K. Beare, 2010. "Copulas and Temporal Dependence," Econometrica, Econometric Society, vol. 78(1), pages 395-410, January.
    3. Beare, Brendan K., 2012. "Archimedean Copulas And Temporal Dependence," Econometric Theory, Cambridge University Press, vol. 28(6), pages 1165-1185, December.
    4. Nummelin, Esa & Tuominen, Pekka, 1982. "Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 12(2), pages 187-202, March.
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