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Convergence rates for empirical measures of Markov chains in dual and Wasserstein distances

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  • Riekert, Adrian

Abstract

We consider a Markov chain on Rd with invariant measure μ. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular the 1-Wasserstein distance. The main result of this article is a new upper bound for the expected distance, which is proved by combining a Fourier expansion with a truncation argument. Our bound matches the known rates for i.i.d. random variables up to logarithmic factors. In addition, we show how concentration inequalities around the mean can be obtained.

Suggested Citation

  • Riekert, Adrian, 2022. "Convergence rates for empirical measures of Markov chains in dual and Wasserstein distances," Statistics & Probability Letters, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:stapro:v:189:y:2022:i:c:s0167715222001468
    DOI: 10.1016/j.spl.2022.109605
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    References listed on IDEAS

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    1. Dedecker, Jérôme & Fan, Xiequan, 2015. "Deviation inequalities for separately Lipschitz functionals of iterated random functions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 60-90.
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