Sufficient MCMC CLT regimes

Put $Y_k=f(X_k)$. This is a centered stationary real sequence by [F1]. It is square-integrable in every case: boundedness suffices in case 1, the power moment suffices in cases 2 and 3 on the probability space, and cases 5 and 6 assume $L^2$. In case 4, split at $\mathrm{\mid }f\mathrm{\mid }=e$: on the bounded part $f^2\le e^2$, and on the other part $f^2\le f^2\log \mathrm{\mid }f\mathrm{\mid }$, so the stated logarithmic moment implies $L^2$. All sequence suppliers below therefore have their centering, stationarity and second-moment premises.

Suppose a pointwise rate $\mathrm{\parallel }{P}^n(x,\cdot )-\pi {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le M(x)a_n$ is available with $\pi (M)<\mathrm{\infty }$. For any past event $A\in \sigma (X_0,\dots ,X_j)$ and separated future event $B\in \sigma (X_{j+n},X_{j+n+1},\dots )$, [F3] supplies $0\le h\le 1$ with $E[{\mathbf{1}}_B\mid \sigma (X_0,\dots ,X_j)]=P^{n}h(X_j)$ and $P(B)=\pi (h)$. Approximating $h$ uniformly by dyadic averages of its superlevel indicators gives $\mathrm{\mid }{P}^{n}h(x)-\pi (h)\mathrm{\mid }\le \mathrm{\parallel }{P}^n(x,\cdot )-\pi {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}$: each indicator discrepancy is bounded by the defining event supremum. Consequently $\mathrm{\mid }P(A\cap B)-P(A)P(B)\mathrm{\mid }=\mathrm{\mid }E[{\mathbf{1}}_A(P^{n}h(X_j)-\pi (h))]\mathrm{\mid }\le \pi (M)a_n\mathrm{.}$ Taking the suprema and then applying [F1] gives ${\alpha }_Y(n)\le {\alpha }_X(n)\le \pi (M)a_n$. No measurability of an uncountable total-variation supremum is needed, because its measurable envelope bounds each event discrepancy pointwise.

In case 1, take $a_n=n^{-m}$ in step 1.2. Since $m>1$, grouping $2^j\le n<2^{j+1}$ bounds each block of ${\sum }_n{n}^{-m}$ by $2^{-j(m-1)}$; [F9] makes the resulting geometric series finite. Thus ${\sum }_n{\alpha }_Y(n)<\mathrm{\infty }$. The bounded branch of [F4] gives absolute covariance summability and the asserted normal limit for the normalized sums of $Y$. This uses the whole approved range $m>1$.

In case 2, put $\theta =\delta \mathrm{/}(2+\delta )$. Step 1.2 gives ${\alpha }_Y(n{)}^{\theta }\le \pi (M{)}^{\theta }{n}^{-m\theta }$. The exact assumption $m\delta >2+\delta$ says $m\theta >1$, so the same dyadic argument and [F9] make these bounds summable. The power-moment branch of [F4] proves the same covariance and distribution conclusions.

In cases 3 and 4, [F5] replaces the possibly nonintegrable original envelope by a finite integrable geometric envelope with some rate $a\in (0,1)$. Thus ${\alpha }_Y(n)\le C{a}^n$ by [F1] and [F5]. In case 3, every positive power of this bound is summable, so the $L^{2+\delta }$ branch of [F4] applies. In case 4, its exponential/logarithmic-moment branch applies directly. Each yields absolute covariance summability and the normal limit without strengthening the given moment assumption.

In case 5, reversibility and the geometric-rate assumptions allow [F6] to give ${\rho }_X(n)\le C{r}^n$ for some $r<1$. By [F1], ${\rho }_Y(n)\le {\rho }_X(n)$, so its series is summable. The $L^2$ rho-mixing theorem in [F6] proves the required covariance-summable CLT.

In case 6, [F7] gives ${\varphi }_X(n)\le C{r}^n$ for some $r<1$. By [F1], ${\sum }_n\sqrt{{\varphi }_Y(n)}\le \sqrt{C}{\sum }_n{r}^{n\mathrm{/}2}<\mathrm{\infty }$. The uniform-mixing $L^2$ theorem in [F7] therefore gives the same conclusions.

In all six cases the covariance series is absolutely convergent, its sum is finite and nonnegative, and the normalized sum converges to the centered normal with precisely that covariance sum. If the sum is zero, [F8] also proves the degenerate limit directly in $L^2$, so no division by an asymptotic variance is made. Stationarity makes ${\sum }_{k=0}^{n-1}{Y}_k$ and ${\sum }_{k=1}^n{Y}_k$ equal in law. The established limit is therefore exactly the displayed limit in the Statement, and only now does [F8] identify its variance as ${\sigma }_f^2$.