Replicated batch means for steady-state simulations with initial transients

We provide asymptotic expressions for the expected value and variance of the replicated batch means variance estimator when the stochastic process being simulated has an additive initial transient. These expressions explicitly show how the initial transient and the autocorrelation in the data affect the performance of the estimator. We apply our results to study how many replications will minimize the asymptotic bias of the variance estimator for a simple example.