Covariance Ordering for Discrete and Continuous Time Markov Chains

The covariance ordering, for discrete and continuous time Markov chains, is defined and studied. This partial ordering gives a necessary and sufficient condition for MCMC estimators to have small asymptotic variance. Connections between this ordering, eigenvalues, and suprema of the spectrum of the Markov transition kernel, are provided. A representation of the asymptotic variance of MCMC estimators in terms of eigenvalues and eigenvectors is extended to continuous time. This representation is used to establish convergence of the asymptotic variance of MCMC estimators derived from the discretization of a continuous time Markov chain.