Mean-square consistency of the variance estimator in steady-state simulation output analysis

In steady-state simulation output analysis, mean-square consistency of the process-variance estimator is important for a number of reasons. One way to construct an asymptotically valid confidence interval around a sample mean is via construction of a consistent estimator of the process variance and a central limit theorem. Also, if an estimator is consistent in the mean-square sense, a mean-square error analysis is theoretically justified. Finally, batch-size selection is an open research problem in steady-state output analysis, and a mean-square error analysis approach has been proposed in the literature; to be valid, the process-variance estimators constructed must be consistent in the mean-square sense. In this paper, we prove mean-square consistency of the process-variance estimator for the methods of batch means, overlapping batch means, standardized time series (area), and spaced batch means, by rigorously computing the rate of decay of the variance of the process-variance estimators. Asymptotic results for third and higher centered moments of the batch means and area variance estimators are also given, along with central limit theorems.