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On Null-Homology and Stationary Sequences

Author

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  • Gerold Alsmeyer

    (University of Münster)

  • Chiranjib Mukherjee

    (University of Münster)

Abstract

The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measure-preserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (Cocycles on ergodic transformation groups. Macmillan lectures in mathematics, vol 1, Macmillan Company of India, Ltd., Delhi, 1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same null-homology equivalence class. We also discuss various aspects of null-homology within the class of Markov random walks and compare null-homology with a formally stronger notion which we call strict-sense null-homology. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.

Suggested Citation

  • Gerold Alsmeyer & Chiranjib Mukherjee, 2023. "On Null-Homology and Stationary Sequences," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-25, March.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-023-01249-6
    DOI: 10.1007/s10959-023-01249-6
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    References listed on IDEAS

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    1. Alsmeyer, Gerold & Buckmann, Fabian, 2019. "An arcsine law for Markov random walks," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 223-239.
    2. Woodroofe, Michael, 1992. "A central limit theorem for functions of a Markov chain with applications to shifts," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 33-44, May.
    3. Gerold Alsmeyer & Fabian Buckmann, 2018. "Fluctuation Theory for Markov Random Walks," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2266-2342, December.
    Full references (including those not matched with items on IDEAS)

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