estimators due to their computational feasibility. Similar construction for multiple-chain output analysis should be possible for the overlapping batch means estimators found in [@fleg:jone:2010; @liu2018batch].

Preliminaries
=============

We present a few preliminary results to assist us in our proofs.

\[lemma: brownian\_bound\] (Csorgo and Revesz (1981)). Suppose Assumption\[ass:bn\] holds, then for all $\epsilon > 0$ and for almost all sample paths, there exists $n_{0}\left(\epsilon\right)$ such that $\forall n\geq n_{0}$ and $\forall i = 1, ..., p$ $$\sup_{0\leq t \leq n-b}\sup_{0 \leq s \leq b} \left| B^{\left(i\right)}\left(t+s\right) - B^{\left(i\right)}\left(t\right) \right| < \left(1+ \epsilon\right)\left(2b\left(\log\dfrac{n}{b} + \log\; \log\; n\right)\right)^{1/2} ,$$ $$\sup_{0 \leq s \leq b} \left|B^{\left(i\right)}\left(n\right) - B^{\left(i\right)}\left(n - s\right)\right| < \left(1+ \epsilon\right)\left(2b\left(\log\dfrac{n}{b} + \log\;\log\;n\right)\right)^{1/2} , \;and$$ $$\left|B^{\left(i\right)}\left(n\right)\right| < \left(1+\epsilon\right)\sqrt{2n\;\log \log n} \; .$$

The following straightforward decomposition will be used often and thus is presented as a lemma.

\[lem:Y\_diff\_simpli\] For $\hat{\mu}_k$ and $\hat{\mu}$ defined as before, $$\sum\limits_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T} = {\displaystyle}\sum_{k=1}^{m} (\hat{\mu}_k - \mu)(\hat{\mu}_k - \mu)^T - m (\hat{\mu} - \mu)(\hat{\mu} - \mu)^T\,.$$

Strong consistency of RBM {#subsec: strong_consistency_RBM}
-------------------------

We will show that the RBM estimator can be decomposed into the ABM estimator, plus some small order terms. Consider $$\begin{aligned}
\hat{\Sigma}_{R,b} &= \dfrac{b}{am-1}\sum_{k=1}^{m}\sum_{s=1}^{a}\left(\bar{Y}_{k,s} - \hat{\mu}\right)\left(\bar{Y}_{k,s} - \hat{\mu}\right)^{T}\\
&= \dfrac{b}{am-1}\sum_{k=1}^{m}\sum_{s=1}^{a}\left(\bar{Y}_{k,s} - \hat{\mu}_k + \hat{\mu}_k - 