Chain CLT from a TV rate: $E_\pi|f|^{2+\delta}<\infty$, $\sum \gamma(n)^{\delta/(2+\delta)}<\infty$ (Jones Cor 1)

Setting. $X$ is an arbitrary type with a measurable-space structure (no countable-generation assumption); $P$ is a kernel from $X$ to $X$ assumed (typeclass) to be a Markov kernel ($P(x,\cdot)$ a probability measure for every $x$); $\pi$ is a measure on $X$ assumed (typeclass) to be a probability measure. Hypotheses. (1) Harris ergodicity: $\pi$ is invariant for $P$ ($A \mapsto \int_X P(x,A)\,d\pi(x)$ equals $\pi$) and for every $x \in X$, $d_{TV}(P^n(x,\cdot),\pi) \to 0$ as $n \to \infty$, where $P^n$ is the $n$-fold iterate ($P^0 =$ identity kernel, $P^{n+1} = P \circ P^n$) and $d_{TV}(\mu,\nu) = \sup\{\,|\mu(A)-\nu(A)| : A \text{ measurable}\,\}$ (no factor $\tfrac12$; measure values sent to reals with $\infty \mapsto 0$; real supremum with junk value $0$ on unbounded sets, and the set contains $0$ via $A = \varnothing$). (2) $f : X \to \mathbb R$ is measurable. (3) $M : X \to \mathbb R$ satisfies $M(x) \ge 0$ for all $x$ and is integrable with respect to $\pi$ ($\pi$-a.e. strongly measurable with $\int |M|\,d\pi < \infty$). (4) $\gamma : \mathbb N \to \mathbb R$ satisfies $\gamma(n) \ge 0$ for all $n$ and is antitone (nonincreasing: $m \le n \Rightarrow \gamma(n) \le \gamma(m)$). (5) Rate hypothesis: for every $x \in X$ and every $n \ge 1$, $d_{TV}(P^n(x,\cdot), \pi) \le M(x)\,\gamma(n)$; nothing is asserted at $n = 0$. (6) $\delta$ is a real number with $0 < \delta$. (7) The function $x \mapsto |f(x)|^{2+\delta}$ is integrable with respect to $\pi$ (real-exponent power of the nonnegative base $|f(x)|$, with $0^{2+\delta} = 0$). (8) The series $\sum_{n=0}^{\infty} \gamma(n)^{\delta/(2+\delta)}$ is summable (real-exponent powers of the nonnegative reals $\gamma(n)$; the exponent $\delta/(2+\delta)$ is strictly between $0$ and $1$; $0^{\delta/(2+\delta)} = 0$; the $n = 0$ term is included in the sum). Conclusion ($f$ satisfies the CLT for $(P,\pi)$, unfolded): there exists a nonnegative real $v$ such that for every probability measure $\lambda$ on $X$ — a single $v$ is claimed to work for all $\lambda$; the existential over $v$ precedes the universal over $\lambda$ — the functions $Z_n(\omega) = \sqrt{n}\,\left(\bar f_n(\omega) - \int_X f\,d\pi\right)$ converge in distribution to $\mathcal N(0,v)$ as $n \to \infty$ under the path measure $\mu_\lambda$. Here $\bar f_n(\omega) = n^{-1}\sum_{i=0}^{n-1} f(\omega_{i+1})$ averages $f$ over the chain states at times $1,\dots,n$ (the time-$0$ state is never used; at $n=0$, Lean's $0^{-1}=0$ and $\sqrt 0 = 0$ give $Z_0 = 0$); the centering constant is the $\pi$-mean $\int f\,d\pi$ for every initial law $\lambda$; $\mu_\lambda$ is the law on $\mathbb N \to X$ of the time-homogeneous Markov chain with transition kernel $P$ started with $\omega_0 \sim \lambda$ (Ionescu–Tulcea construction, $\omega_{n+1} \sim P(\omega_n,\cdot)$); convergence in distribution means $\int g(Z_n)\,d\mu_\lambda \to \int_{\mathbb R} g\,d\mathcal N(0,v)$ for every bounded continuous $g : \mathbb R \to \mathbb R$; and $\mathcal N(0,v)$ is the normal law of mean $0$ and variance $v$, equal to the Dirac point mass at $0$ when $v = 0$, so a degenerate limit is allowed.