Convergence properties of the batch means method for simulation output analysis

We examine key convergence properties of the steady-state simulation analysis method of nonoverlapping batch means (NOBM) when it is applied to a stationary, phi-mixing process. For an increasing batch size and a fixed batch count, we show that the standardized vector of batch means converges in distribution to a vector of independent standard normal variates – a well-known result underlying the NOBM method for which there appears to be no direct, readily accessible justification. To characterize the asymptotic behavior of the classical NOBM confidence interval for the mean response, we formulate certain moment conditions on the components (numerator and denominator) of the associated Student's t-ratio that are necessary to ensure the validity of the confidence interval. For six selected stochastic systems, we summarize an extensive numerical analysis of the convergence to steady-state limits of these moment conditions; and for two systems we present a simulation-based analysis exemplifying the correspnding convergence in distribution of the components of the NOBM t-ratio. These results suggest that in many simulation output processes, approximate joint normality of the batch means is achieved at a substantially smaller batch size than is required to achieve approximate independence; and an improved batch means method should exploit this property whenever possible.