Conditions of asymptotic normality of one-step $M$-estimators

In the case of independent identically distributed observations we study asymptotic behavior of one-step $M$-estimators which are explicit approximations to the corresponding consistent $M$-estimators. In particulary, we find quite general conditions for asymptotic normality of one-step $M$-estimators under consideration. As a consequence, we consider Fisher's one-step approximations to consistent maximum likelihood estimators.