Are the mean of samples taken from Metropolis-Hastings MCMC normally distributed? 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 $\phi : \mathbb X \to \mathbb R^n$.
 A: Indeed when the univariate Markov chain central limit theorem holds, the multivariate CLT also holds. In fact, the the univariate CLT is a consequence of the multivariate CLT via the Cramer-Wold Theorem. Look at Page 7-9 here.
However, the Theorem 4.4 you indicated has the hidden line of "satisfying enough regularity conditions". These regularity conditions are difficult to verify but do hold in general. These conditions are that of the rate of convergence of the Markov chain to its stationary distribution. This paper summarizes most of the research done in this field. For the Metropolis-Hasting MCMC (which is reversible), you need that the chain be geometrically ergodic with a finite 2nd moment.
If these conditions hold true, the the multivariate CLT takes the following form: If $\phi: \mathbb{X} \to \mathbb{R}^n$, and $E_{\pi} \|\phi(X_1) \|^2 <\infty$ and $\{X_i\}$ is geometrically ergodic, then there exists a $n \times n$ positive definite matrix, $\Sigma$ such that,
$$\sqrt{t} \left[\dfrac{1}{t}\sum_{i=1}^t{ \phi(X_i)} - \int_\mathbb{X} \phi(x) \pi(x) \right] \to N_n(0, \Sigma), $$
where if $\phi(X_t) = Y_t = (Y^{(1)}_t, Y^{(2)}_t, \dots, Y^{(n)}_t)$, then the $(i,j)th$ entry of $\Sigma$ is
$$ \Sigma_{ij} = Cov_{\pi}(Y^{(i)}_1, Y^{(j)}_1) + 2 \sum_{k=1}^{\infty} Cov_{\pi}(Y^{(i)}_1, Y^{(j)}_{1+k}). $$
