Ordering Monte Carlo Markov Chains

Markov chains having the same stationary distribution can be partially ordered by performance in the central limit theorem. We say that one chain is at least as good as another in the e ciency partial ordering if the variance in the central limit theorem is at least as small for every L( ) functional of the chain. Peskun partial ordering implies e ciency partial ordering [25, 30]. Here we show that Peskun partial ordering implies, for nite state spaces, ordering of all the eigenvalues of the transition matrices, and, for general state spaces, ordering of the suprema of the spectra of the transition operators. We also de ne a covariance partial ordering based on lag one autocovariances and show that it is equivalent to the e ciency partial ordering when restricted to reversible Markov chains. Similar but weaker results are provided for non-reversible Markov chains.