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How to proof that every positive Harris recurrent, aperiodic, stationary Markov chain is alpha-mixing (strong-mixing)?

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In the following, all results are referenced from Bradley (2005).

Suppose that $X := (X_k, k \in \mathbb{Z})$ is a strictly stationary Markov chain. If $X$ is Harris recurrent and aperiodic then $X$ is $\beta$-mixing (Theorem 3.5), and hence $\alpha$-mixing (Section 2.1 or Theorem 3.2).

References

Bradley, Richard C. (2005), Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions. Probability Surveys 2, 107-144. doi: 10.1214/154957805100000104

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