  Autocovariances are a fundamental quantity of interest in Markov chain Monte
Carlo (MCMC) simulations with autocorrelation function (ACF) plots being an
integral visualization tool for performance assessment. Unfortunately, for
slow-mixing Markov chains, the empirical autocovariance can highly
underestimate the truth. For multiple-chain MCMC sampling, we propose a
globally-centered estimator of the autocovariance function (G-ACvF) that
exhibits significant theoretical and empirical improvements. We show that the
bias of the G-ACvF estimator is smaller than the bias of the current
state-of-the-art. The impact of this improved estimator is evident in three
critical output analysis applications: (1) ACF plots, (2) estimates of the
Monte Carlo asymptotic covariance matrix, and (3) estimates of the effective
sample size. Under weak conditions, we establish strong consistency of our
improved asymptotic covariance estimator, and obtain its large-sample bias and
variance. The performance of the new estimators is demonstrated through various
examples.
