is on MCMC, but our results apply more generally to $\alpha$-mixing processes. Our first assumption is on the mixing rate of $\{X_{k,t}\}$ and the moments of $\{Y_{k,t}\}  = \{h(X_{k,t})\}$.

\[ass:mixing\] For some $\delta > 0$, ${\text{E}}_F\|Y_{k,1}\|^{2 + \delta} < \infty$ and there exists $\epsilon > 0$ such that $\{X_{k,t}\}$ is $\alpha$-mixing with $\alpha(n) = o\left(n^{-\left(2 + \epsilon\right)\left(1 + 2/\delta\right)}\right)$ for all $k = 1, \dots, m$.

Assumption \[ass:mixing\] will be our umbrella assumption on the process, however, we bring special focus to when this assumption is satisfied by Markov chains. Let $P: {\mathcal{X}}\times {\mathcal{B}}({\mathcal{X}}) \to [0,1]$ be an $F$-invariant Markov chain transition kernel, where for $x \in {\mathcal{X}}$ and $A \in {\mathcal{B}}({\mathcal{X}})$, $P(x, A) = \Pr(X_2 \in A |X_1 = x)$. The $n$-step transition is $P^n(x,A) = \Pr(X_{n+1} \in A \mid X_1 = x)$. We assume that $P$ is Harris ergodic, which guarantees ergodicity and implies that the Markov chain is $\alpha$-mixing [@jone:2004]. Assumption \[ass:mixing\] may hold when the Markov chain is *polynomially ergodic*. That is, suppose there exists $M:{\mathcal{X}}\to [0, \infty)$ with ${\text{E}}_F M < \infty$ and $\xi > 0$ such that $$\|P^n(x,\cdot) - F(\cdot) \|_{TV} \leq M(x) n^{-\xi}\,,$$ where $\|\cdot\|_{TV}$ is the total variation norm. Polynomially ergodic Markov 