On the batch means and area variance estimators

In steady-state simulation output analysis, construction of a consistent estimator of the the variance parameter of the process may be desirable in a number of instances. For example, if the process obeys a central limit theorem and an estimator of the variance is available, one may, then, construct an asymptotically valid confidence interval for the process-mean parameter. Centered moments (e.g. bias, variance, skewness, etc.) of an estimator are familiar measures of goodness of that estimator. Also, a central limit theorem involving the estimator provides its asymptotic rate of convergence. We consider here the batch means and the standardized time series area variance estimators, in their nonclassical setting, and provide asymptotic expressions for their centered moments as well as central limit theorems. As a by-product, consistency in the mean-square sense of these estimators is obtained. Our assumption on the process does not include stationarity nor covariance stationarity (although we are in the steady-state context).