and variance results for the larger class of $\alpha$-mixing processes. Since deviation is measured from the global mean, in the event that chains explore different parts of the state space, RBM estimates yield dramatic improvements over ABM. Asymptotically, RBM and ABM are no different, but RBM has clear advantages over ABM in finite samples which we illustrate through three examples: a Gibbs sampler for a bivariate Gaussian target, a Metropolis-Hastings sampler for the tricky Rosenbrock distribution, and a Metropolis-Hastings sampler for a Bayesian multinomial regression posterior. For the Gaussian Gibbs sampler, we obtain the true value of $\Sigma$. In the past, the true $\Sigma$ has only been known for some time-series processes, but never for a more traditional MCMC problem. Obtaining the true $\Sigma$ allows for this example to be used for benchmarking the performance of new estimators in the future. Over all three examples, the results are consistent: the RBM estimator is at least as good as the ABM estimator with no additional computation costs. However, when dealing with slow mixing Markov chains or small sample sizes, RBM can far outperform ABM, yielding a significantly more reliable estimator of $\Sigma$.

Background and assumptions {#sec:background_and_assumptions}
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Much of our focus 