the style of [@glyn:whit:1992] lead to asymptotically valid confidence regions. Specifically for MCMC, Theorem \[thm:RBM\_consistency\] allows the usage of effective sample size as a valid termination rule. We will discuss the impact of this in Section \[sec:examples\].

Let $\hat{\Sigma}^{ij}_{R,L}$ and $\Sigma_{ij}$ be the $ij$th element of $\hat{\Sigma}_{R,L}$ and $\Sigma$, respectively. For any $k = 1, \dots, m$, define $$\Gamma = - \sum_{s=1}^{\infty}s\left[{\text{Cov}}_{F}\left(Y_{k,1}, Y_{k,1+s}\right) + {\text{Cov}}_{F}\left(Y_{k,1}, Y_{k,1+s}\right)^{T}\right]\,.$$

Our next results establishes the large-sample bias and variance of the lugsail RBM estimators. The proofs of the following two theorems are in Appendix \[subsec:bias\_RBM\] and Appendix \[subsec:variance\_RBM\].

\[thm:rbm\_bias\] If Assumption \[ass:mixing\] holds, $$\text{Bias}\left(\hat{\Sigma}_{R,b} \right) = \left(\dfrac{1-rc}{1-c} \right)\dfrac{\Gamma}{b} + o\left(\dfrac{1}{b}\right) \; .$$

\[thm:rbm \_variance\] If Assumptions \[ass:mixing\] and \[ass:bn\] hold, ${\text{E}}D^{4} < \infty$, $E_{F}\|Y_{1,1}\|^{4} < \infty$, and $b^{-1}\psi^{2}(n)\log n \to 0$ as $n \to \infty$, $$\dfrac{n}{b}Var\left(\hat{\Sigma}^{ij}_{R,L}\right) = \dfrac{1}{m}\left[\left(\dfrac{1}{r} + \dfrac{r - 1}{r\left(1 - c\right)^{2}}\right)\left(\Sigma^{2}_{ij} + \Sigma_{ii}\Sigma_{jj}\right)\right] + o\left(1\right) \; .$$

These results mimic those of the ABM estimator (Section \[sec:abm\]) and the single-chain BM estimator. The results together ensure means-square consistency of the lugsail RBM estimator.

Other multiple chain estimators {#sec:other}
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### Average batch means {#sec:abm}

A natural way to combine variance estimators from $m$ chains is to average them, which yields the 