Many applications in the field of statistics require Markov chain Monte Carlo
methods. Determining appropriate starting values and run lengths can be both
analytically and empirically challenging. A desire to overcome these problems
has led to the development of exact, or perfect, sampling algorithms which
convert a Markov chain into an algorithm that produces i.i.d.\ samples from the
stationary distribution. Unfortunately, very few of these algorithms have been
developed for the intractable distributions that arise in statistical
applications, which typically have uncountable support.
Calculating a Monte Carlo standard error (MCSE) is an important step in the
statistical analysis of the simulation output obtained from a Markov chain
Monte Carlo experiment. For example, it can be used to provide a rigorous
method for terminating the simulation. An MCSE is usually based on an estimate
of the variance of the asymptotic normal distribution. We consider spectral and
batch means methods for estimating this variance. In particular, we establish
conditions which guarantee that these estimators are strongly consistent as the