Strong Consistency of the Variance Estimator in Steady-State Simulation Output Analysis

In this paper, we study the problem of variance estimation in steady-state simulation output analysis. A confidence interval for the mean of an output sequence may be constructed using a (weakly) consistent estimator of the time-average variance constant of the process. Moreover, a strongly consistent estimator of the variance constant is required for fixed-width confidence-interval estimation procedures to be asymptotically valid. Here, we establish sufficient conditions to ensure strong consistency of the estimator of the variance constant obtained by several widely used methods of steady-state simulation output analysis, namely, batch means, overlapping batch means, spaced batch means, and standardized time series (area estimator). We also introduce the overlapping area estimator of the variance constant. The established sufficient conditions provide qualitative insight into the relation between the correlation of the process and certain parameters (e.g., batch size) of the output method used.