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Let $X_1, X_2, \dots, X_n$ be $n$ samples from a discrete-time continuous-space Markov Chain.

Are there any good references who have provided a Chernoff-type bound regarding the behaviour of the ergodic mean $\bar X = \frac 1 n \sum_{i = 1}^n X_i$ for this type of chain?

There is a good line of work about finite-state chains (e.g. http://www.columbia.edu/~khl2114/files/CLLM.pdf, https://arxiv.org/abs/1907.04467) but I cannot find any work related to continuous-space (i.e. infinite-space) Markov Chains.

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    $\begingroup$ The contraction-based results continue to hold for continuous state spaces (though the Doeblin contraction condition is rather strong): arxiv.org/abs/math/0610427 $\endgroup$
    – Aryeh Kontorovich
    Commented Sep 6, 2020 at 8:18
  • $\begingroup$ Thank you very much for your answer! My hunch was similar but I wanted to see it formalized somewhere. $\endgroup$
    – bolzano
    Commented Sep 8, 2020 at 15:58

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