Simulation has been successfully used for estimating performance measures (e.g. mean, variance and quantiles) of complex systems, such as queueing and inventory systems. However, parameter estimation using simulation may be a difficult task under some conditions. In this paper, we present a counterexample for which traditional simulation methods do not allow us to estimate the accuracy of the point estimators for the mean and risk performance measures for steady-state. The counterexample is based on a Markov chain with continuous state space and non-geometric ergodicity. The simulation of this Markov chain shows that neither multiple replications nor batch-based methodologies can produce asymptotically valid confidence intervals for the point estimators. © 2013 AIP Publishing LLC.