Strong consistency of multivariate spectral variance estimators in Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. TheMonte 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.