  Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of
interest of a distribution. The Monte Carlo error in estimation has an
asymptotic normal distribution whose multivariate nature has so far been
ignored in the MCMC community. We present a class of multivariate spectral
variance estimators for the asymptotic covariance matrix in the Markov chain
central limit theorem and provide conditions for strong consistency. We examine
the finite sample properties of the multivariate spectral variance estimators
and its eigenvalues in the context of a vector autoregressive process of order
1.
