$\alpha(n) \le \gamma(n)\, E_\pi M$ from a total-variation rate (Jones Thm 2(ii))

Let $X$ be a Markov chain with transition kernel $P$, Harris ergodic with invariant probability $\pi$, and suppose the total-variation rate bound $\|P^n(x, \cdot) - \pi\| \le M(x)\, \gamma(n)$ holds for all $x$ and all $n \ge 1$, where $M \ge 0$ is integrable with respect to $\pi$ and $\gamma \ge 0$. Then the strong mixing coefficients of the stationary chain satisfy

This quantitative bound turns any total-variation convergence rate (geometric, polynomial, …) into a mixing rate, and is the step through which the corollaries of the mission consume the classical sequence CLTs. The source records the consequence $\alpha(n) = O(\gamma(n))$; the sharp inequality stated here is what its derivation gives.

Formalization Note "Harris ergodic" is encoded by its total-variation characterization: $\pi$ is invariant for $P$ and $\|P^n(x, \cdot) - \pi\| \to 0$ for every starting point $x$ (equivalent to the classical aperiodic, $\psi$-irreducible, positive Harris recurrent definition; the "every $x$" quantifier is exactly the Harris property). Convergence in distribution is weak convergence of laws, and $N(0, 0)$ is read as the point mass at $0$, which absorbs the source's "$\sigma_f^2 > 0$" caveat.