Efficiency of finite state space Monte Carlo Markov chains

The class of finite state space Markov chains, stationary with respect to a common pre-specified distribution, is considered. An easy-to-check partial ordering is defined on this class. The ordering provides a sufficient condition for the dominating Markov chain to be more efficient. Efficiency is measured by the asymptotic variance of the estimator of the integral of a specific function with respect to the stationary distribution of the chains. A class of transformations that, when applied to a transition matrix, preserves its stationary distribution and improves its efficiency is defined and studied.