Large-sample results for batch means

In analyzing the output process generated by a steady-state simulation, we often seek to estimate the expected value of the output. The sample mean based on a finite sample of size n is usually the estimator of choice for the steady-state mean; and a measure of the sample mean's precision is the variance parameter, i.e. the limiting value of the sample size multiplied by the variance of the sample mean as n becomes large. This paper establishes asymptotic properties of the conventional batch-means (BM) estimator of the variance parameter as both the batch size and the number of batches become large. In particular, we show that the BM variance estimator is asymptotically unbiased and convergent in mean square. We also provide asymptotic expressions for the variance of the BM variance estimator. Exact and empirical examples illustrate our findings.