Investigated are the small sample behavior and convergence properties of confidence interval estimators (CIE's) for the mean of a stationary discrete process. We consider CIE's arising from nonoverlapping batch means, overlapping batch means, and standardized time series, all of which are commonly used in discrete-event simulation. For a specific CIE, the performance measures of interest include the coverage probability, and the expected value and variance of the half-length. We use both empirical and analytical methods to make detailed comparisons regarding the behavior of the CIE's for a variety of stochastic processes. All of the CIE's under study are asymptotically valid; however, they are usually invalid for small sample sizes. We find that for small samples, the bias of the variance parameter estimator figures significantly in CIE coverage performance-the less bias the better. A Secondary role is played by the Marginal distribution of the stationary process. Not all CIE's are equal - some require fewer observations before manifesting the properties for CIE validity.