# positive Harris recurrent, aperiodic, stationary Markov chain

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

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).