Geometrically ergodic chains are exponentially strongly mixing: $\alpha(n) \le c\,a^n$

Let $X = \{X_n\}_{n \ge 0}$ be a Markov chain with transition kernel $P$ on a state space $\mathsf{X}$, Harris ergodic with invariant probability distribution $\pi$, and suppose the chain is geometrically ergodic: there are a function $M \ge 0$ and a constant $t < 1$ with

Then the stationary chain (started from $X_0 \sim \pi$) is exponentially fast strongly mixing: there exist constants $c \ge 0$ and $a \in [0, 1)$ such that its strong mixing coefficients satisfy

This is the sentence of the source that the geometric cases of Corollaries 2 and 3 actually consume: "geometrically ergodic Markov chains enjoy exponentially fast strong mixing" (Jones 2004, §3, immediately after the coupling bound $\alpha(n) \le \gamma(n) E_\pi M$ of Theorem 2(ii)). It is what Theorem 6 (Doukhan–Massart–Rio) and Theorem 5(ii) (Ibragimov–Linnik) need as input, and it replaces the intermediate claim that the rate constant $M$ itself may be taken $\pi$-integrable — a claim which is false on general state spaces (the countdown chain on a countable–cocountable space, disproved on this platform as geometricallyErgodic_integrable_rate) even though its mixing consequence survives.

Why the statement holds on an arbitrary measurable space. The countdown counterexample only refutes integrability of the given constant: there $\pi$ is a point mass, the stationary chain is constant, and every $\alpha(n)$ is $0$. The general statement reduces to the countably generated case. Each event in the past and future $\sigma$-algebras of the coordinate process is measurable for the process $\sigma$-algebra generated by countably many measurable subsets of $\mathsf{X}$, and a single countable family realises every $\alpha(n)$ up to a factor $2$ simultaneously (exists_countable_alpha_witnesses). Closing that family under finite Boolean operations and under the maps $A \mapsto \{x : P(x, A) > q\}$, $q \in \mathbb{Q}$, yields a countably generated sub-$\sigma$-algebra $\mathcal{G}$ on which $P$ is again a Markov kernel; $\pi|_{\mathcal{G}}$ is invariant, total variation over $\mathcal{G}$-sets is dominated by total variation over all sets, so Harris ergodicity and geometric ergodicity (same $M$, $t$) pass to the restricted chain, and its path law is the restriction of chainMeasure P π to the $\mathcal{G}$-cylinders. The countably generated case is the classical Meyn–Tweedie route: geometric ergodicity yields a geometric drift function $V \ge 1$ towards a small set (Theorem 15.0.1), $E_\pi V < \infty$ (Theorem 14.3.7), a total-variation rate with constant proportional to $V$ (Remark 1 of the source), and finally $\alpha(n) \le t^n E_\pi V$ by Theorem 2(ii).

Formalization Note "Harris ergodic" is the mission's total-variation encoding (HarrisErgodic: $\pi$ invariant and $\|P^n(x,\cdot) - \pi\| \to 0$ for every $x$), and geometric ergodicity is GeometricallyErgodic from Def_MarkovErgodicity, with a pointwise finite but not necessarily measurable or integrable constant $M$. The mixing coefficient is alphaMixingCoef of the coordinate process $\omega \mapsto \omega_i$ under the stationary path law chainMeasure P π, the same object bounded by alpha_mixing_le_tv_rate. The bound is asked for at every lag including $n = 0$, which costs nothing since $\alpha(0) \le 1/4$. No countably generated hypothesis is imposed, so the statement can be cited by Corollaries 2 and 3 as formalized.