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
efficiency partial ordering if the variance in the
central limit theorem is at least as small for every
functional of the chain. Peskun partial ordering implies efficiency
partial ordering (Peskun, 1973; Tierney, 1995).
Here we show that Peskun partial ordering implies, for finite 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 define a covariance partial ordering based on lag one autocovariances and show that it is equivalent to the efficiency partial ordering when restricted to reversible Markov chains. Similar but weaker results are provided for non-reversible Markov chains.