Strong Consistency and Other Properties of the Spectral Variance Estimator

Consistent estimation of the variance parameter of a stochastic process allows construction, under certain conditions, of a confidence interval for the mean of the process. If the variance estimator is strongly consistent, fixed-width confidence interval construction is valid for large samples. It has long been known that the spectral variance estimator of steady-state simulation output analysis is consistent in the mean-square sense. Here, we provide strong consistency of this estimator, thereby justifying fixed-width confidence interval construction for the spectral method. A characterization of spectral density function estimators is also introduced. This characterization provides insight into the relation between spectral methods and overlapping batch means-type variance estimators. Finally, some of the mathematical conditions provide qualitative insight into the relation between the process correlation and certain parameters of spectral methods.