The information in the marginal law of a Markov chain

If we have a parametric model for the invariant distribution of a Markov ch ain but cannot or do not want to use any information about the transition d istribution (except, perhaps, that the chain is reversible), what is the be st use we can make of the observations? We determine a lower bound for the asymptotic variance of regular estimators and show constructively that the bound is attainable. The results apply to discretely observed diffusions.