Let $(X_n)_{n \in \mathbb N}$ be a stationary, aperiodic, irreducible, finite state space Markov chain. Define the $\alpha$-mixing coefficient as:
$$\alpha(n) = \sup \{\vert \Pr(A \cap B) - \Pr(A)\Pr(B)\vert\mid|t-s| \geq n, A \in \mathscr{F}_{\leq t}, B\in \mathscr{F}_{\geq s}\}$$
where $\mathscr{F}_{\leq t}$ is the sigma-algebra generated by $(X_u)_{u\le t}$ and $\mathscr{F}_{\geq s}$ the sigma-algebra generated by $(X_u)_{u\ge s}$.
A result which is commonly stated in many papers is the following:
$$ \alpha(n) \to 0, \qquad n \to \infty$$
and in particular the decay is exponentially fast. I have not been able to find a simple proof of the exponential decay though. Is there a straightforward explanation or any reference which shows this?
An example paper which makes the above claim is the following:
Bradley, Richard C. "Basic properties of strong mixing conditions. A survey and some open questions." Probability surveys 2.2 (2005): 107-144.