I've come across the following theorem while studying MCMC. It seems to suggest that the sample mean taken from the MCMC – the posterior marginal expectation – should be normally distributed, using just $\phi(x) = x$:
Theorem 4.4. (A Central Limit Theorem). For a Harris recurrent, $\pi$-invariant Markov chain, and a function $\phi : \mathbb X \to \mathbb R$ satisfying enough regularity conditions, $$ \sqrt{t} \left[ \frac1t \sum_{i=1}^t \phi(X_t) - \int_{\mathbb X} \, \phi(x) \pi(x) \, dx \right] \mathop{\longrightarrow}\limits_{t \to \infty}^D \mathcal N(0, \sigma^2(\phi)) $$ where $$ \sigma^2(\phi) = \mathbb V[\phi(X_1)] + 2 \sum_{k=2}^\infty \mathbb{Cov}[\phi(X_1), \phi(X_k)] .$$ The variance and covariance in the expression above are with respect to the distribution $\pi$ of the Markov chain in its stationary regime.
I am wondering is there some extension of this result when dimension of the parameter space is larger than 1? So in this case $\phi : \mathbb X \to \mathbb R^n$.