combined estimator of $\mu$ from the $m$ independent chains is $\hat{\mu} = m^{-1} \sum_{k=1}^{m} \hat{\mu}_k\,.$ If a Markov chain central limit theorem (CLT) holds, then irrespective of the starting value, there exists a $p \times p$ positive-definite matrix $\Sigma$ such that for all $k = 1, \dots, m$, as $n \to \infty$, $$\label{eq:multi_CLT_each}
\sqrt{n}(\hat{\mu}_k - \mu) \overset{d}{\to} N_p(0, \Sigma)\,.$$ Consequently, a CLT holds for $\hat{\mu}$ as well so that as $n \to \infty$, $$\label{eq:multi_CLT}
\sqrt{mn}(\hat{\mu} - \mu) \overset{d}{\to} N_p(0, \Sigma)\,.$$ A significant goal in MCMC output analysis is estimating $\Sigma$ in order to ascertain the quality of estimation of $\mu$ via $\hat{\mu}$. This has a direct consequence on determining when to stop the MCMC sampler [@bro:gel:1998; @fleg:hara:jone:2008; @gelm:rubi:1992a; @gong:fleg:2016; @roy:2019; @vats:fleg:jon:2019; @vehtari2020rank]. Additionally, estimators of $\Sigma$ must be (strongly) consistent in order to yield nominal coverage for large termination sizes. Amongst the list of estimators, only the naive estimator of [@broo:robe:1998], which estimates $\Sigma$ by the sample covariance matrix of $\hat{\mu}_1, \dots, \hat{\mu}_m$, is specifically constructed for multiple-chains. We show that this estimator is not consistent and exhibits high variability when the number of parallel implementations is small.

Many estimators have been introduced for a single-chain implementation, including batch means 