Reversible Markov chains and optimality of symmetrized empirical estimators

Suppose that we want to estimate the expectation of a function of two argum ents under the stationary distribution of two successive observations of a reversible Markov chain. Then the usual empirical estimator can be improved by symmetrizing. We show that the symmetrized estimator is efficient. We p oint out applications to discretely observed continuous-time processes. The proof is based on a result for general Markov chain models which can be us ed to characterize efficient estimators in any model defined by restriction s on the stationary distribution of a single or two successive observations .