  Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating
expectations with respect to a target distribution. A fundamental question is
when should sampling stop so that we have good estimates of the desired
quantities? The key to answering this question lies in assessing the Monte
Carlo error through a multivariate Markov chain central limit theorem (CLT).
The multivariate nature of this Monte Carlo error largely has been ignored in
the MCMC literature. We present a multivariate framework for terminating
simulation in MCMC. We define a multivariate effective sample size, estimating
which requires strongly consistent estimators of the covariance matrix in the
Markov chain CLT; a property we show for the multivariate batch means
estimator. We then provide a lower bound on the number of minimum effective
samples required for a desired level of precision. This lower bound depends on
the problem only in the dimension of the expectation being estimated, and not
on the underlying stochastic process. This result is obtained by drawing a
connection between terminating simulation via effective sample size and
terminating simulation using a relative standard deviation fixed-volume
sequential stopping rule; which we demonstrate is an asymptotically valid
procedure. The finite sample properties of the proposed method are demonstrated
in a variety of examples.
