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Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains

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  • Alain Durmus

    (École polytechnique)

  • Eric Moulines

    (École polytechnique)

  • Alexey Naumov

    (HSE University)

  • Sergey Samsonov

    (HSE University)

Abstract

In this paper, we establish moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in weighted total variation norm or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider.

Suggested Citation

  • Alain Durmus & Eric Moulines & Alexey Naumov & Sergey Samsonov, 2024. "Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2184-2233, September.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:3:d:10.1007_s10959-024-01315-7
    DOI: 10.1007/s10959-024-01315-7
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    References listed on IDEAS

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    1. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
    2. Beskos, A. & Pinski, F.J. & Sanz-Serna, J.M. & Stuart, A.M., 2011. "Hybrid Monte Carlo on Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2201-2230, October.
    3. Butkovsky, O.A. & Veretennikov, A.Yu., 2013. "On asymptotics for Vaserstein coupling of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3518-3541.
    4. Doukhan, Paul & Neumann, Michael H., 2007. "Probability and moment inequalities for sums of weakly dependent random variables, with applications," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 878-903, July.
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