is, $$\hat{\Sigma}_{N} := \dfrac{n}{m-1}\sum_{k=1}^{m}\left(\hat{\mu}_k - \hat{\mu}\right)\left(\hat{\mu}_k - \hat{\mu}\right)^{T}\;.$$ In principle, $\hat{\Sigma}_{N}$ would perform well if the number of parallel runs, $m$, was reasonably large. In fact, if the asymptotics are in both $m$ and $n$, $\hat{\Sigma}_N$ is a viable estimator. However, in most situations, the number of parallel implementations of a Markov chain is small, and the naive estimator has a high variance even for large $n$. The following theorem shows that $\hat{\Sigma}_N$ is not consistent for $\Sigma$ for fixed $m$; the proof is in Appendix \[subsec: naive\_convergence\].

\[thm:naive\_convergence\] Under Assumption \[ass:mixing\], $\hat{\Sigma}_N \overset{d}{\to} (m-1)^{-1} \text{Wishart}_p(\Sigma, m-1)$ as $n \to \infty$.

We will demonstrate the instability of $\hat{\Sigma}_N$ in an example where the true $\Sigma$ is known, mimicking what has been witnessed by [@fleg:hara:jone:2008].

Examples {#sec:examples}
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In each example, we compare the performance of RBM against ABM and the naive estimator. We implement lugsail versions of ABM and RBM with $r = 3$ and $c = 1/2$, as recommended by [@vats:flegal:2018]. Over repeated simulations, we assess coverage probabilities of confidence regions for $\mu$ when the truth is available, and also focus on running plots of the Frobenius norm, $\|\hat{\Sigma}\|_{F}$ and estimated effective sample size. For slow mixing Markov chains, 