  Markov chain Monte Carlo is a method of producing a correlated sample in
order to estimate features of a target distribution via ergodic averages. A
fundamental question is when should sampling stop? That is, when are the
ergodic averages good estimates of the desired quantities? We consider a method
that stops the simulation when the width of a confidence interval based on an
ergodic average is less than a user-specified value. Hence calculating a Monte
Carlo standard error is a critical step in assessing the simulation output. We
consider the regenerative simulation and batch means methods of estimating the
variance of the asymptotic normal distribution. We give sufficient conditions
for the strong consistency of both methods and investigate their finite sample
properties in a variety of examples.
