chains of order $\xi > \left(2 + \epsilon\right)\left(1 + 2/\delta\right)$ satisfy Assumption \[ass:mixing\] [@jone:2004].

Single chain batch means estimator {#sub:single_chain_batch_means_estimator}
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Let $n = ab$, where both $a,b \in \mathbb{N}$, $b$ is the size of each batch, and $a$ is the number of batches. The values of $a$ and $b$ are chosen so that they satisfy the following assumptions.

\[ass:bn\] The integer sequence $b$ is such that $b \to \infty$ and $b/n \to 0$ as $n \to \infty$, and both $b$ and $n/b$ are non decreasing.

Additional assumptions on $a$ and $b$ may be needed, but Assumption \[ass:bn\] is necessary for (weak) consistency and mean-square consistency of BM estimators [@glyn:whit:1991; @damerdji:1995]. Let $\bar{Y}_{k,l}$ denote the mean vector for the $l$th batch in the $k$th chain. That is, $$\bar{Y}_{k,l} = \dfrac{1}{b} {\displaystyle}\sum_{t=1}^{b} Y_{k,(l-1)b + t} \qquad l = 1, \dots, a\,.$$ For the $k$th chain, [@chen:seila:1987] define the multivariate BM estimator as $$\label{eq:singe_bm}
    \hat{\Sigma}_{k,b} = \dfrac{b}{a - 1}\sum_{l = 1}^{a}\left(\bar{Y}_{k,l} - \hat{\mu}_k\right)\left(\bar{Y}_{k,l} - \hat{\mu}_k\right)^{T} \,.$$

BM estimators have been used extensively in steady-state simulation and MCMC as they are computationally adept at handling the typical high-dimensional output generated by simulations. For slow mixing processes, the traditional BM estimator is inclined to 