= and

J ((dQ )1/2 )2 Q(x,dy) dQ(x,y) -1-~(T-iJ)c'(x,y) ::; (T-!9) 2rT(x) (2.9)

Q(x, dy) = <p(y-1'Jx) dx, where <pis the standard normal density. If 13 < I, then the chain is ergodic, and Hellinger differentiability holds with c'(x,y) = x(y -1'Jx). Hence (Dr)(x,y) = rx(y -1'Jx), the Fisher information is

3. ASVMPTOTICALL Y LINEAR AND REGULAR ESTIMATORS As in the previous Section, we consider a general Markov chain model with possibly infinite-dimensional parameter. We now want to estimate a onedimensional function of the parameter. We introduce a class of estimators, the "asymptotically linear" estimators, whose properties are particularly easy to study. Our aim is to find an optimal estimator in this class. We show that this problem has a meaningful answer if we further restrict attention to "regular" estimators. The restriction to asymptotically linear estimators is justified by the characterization in Eq. (5.2) of efficient estimators.