The information in the marginal law of a Markov chain

Abstract

If we have a parametric model for the invariant distribution of a Markov chain but cannot or do not want to use any information about the transition distribution (except, perhaps, that the chain is reversible), what is the best 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.