Suppose we use MCMC to estimate:
$$ \mathbf{E}_\pi (h) \approx \frac{1}{N}\sum_{i=1}^{N} h(X_i) $$
If a Markov chain is geometrically ergodic and there is some $ \delta > 0 $ such that $ \mathbf{E}_\pi( |h|^{2+\delta}) < \infty $, then the CLT applies and we can compute the standard error.
What if the CLT does not apply? Is it still possible to compute the standard error of the MCMC estimate? And if so, how?
Let $$ \hat{\mu}_h:= \dfrac{1}{N} \sum_{t=1}^{N} h(X_t)\,. $$
Suppose a CLT does not hold for $\hat{\mu}_h$. In order to estimate the standard error, it must be finite. That is, even though a CLT does not hold $$ \sigma^2_h:=\lim_{N \to \infty} N\text{Var}_{\pi}(\hat{\mu}_h) < \infty. $$
As long as we are willing to assume $\sigma^2_h < \infty$, then by Kipnis and Vardhan(1986), Theorem 1.1, for reversible Markov chains $$ \dfrac{1}{\sigma^2_h \sqrt{N}} (N \hat{\mu}_g - \mu) \overset{d}{\to} N(0, 1)\,. $$
In order to construct confidence intervals etc, $\sigma^2_h$ must be estimated, and in order to ensure normality in the limit, a consistent estimator of $\sigma^2_h$ is required. Unfortunately, consistency of estimators of $\sigma^2_h$ typically requires geometric or polynomial ergodicity ( Jones et. al (2006), Vats et. al. (2019) ).
Thus, in order to estimate $\sigma^2_h$ consistently, we will typically need to make these stronger assumptions on the process. However, one can used "fixed batch" estimators which are not consistent, in which case the asymptotic distribution is non-normal, but one exists. This argument can be found in detail in Atchade(2014).
