Asymptotic properties of one-step $M$-estimators based on nonidentically distributed observations

We study asymptotic behavior of one-step $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent $M$-estimators. These estimators generalize Fisher's one-step approximations to consistent maximum likelihood estimators. Sufficient conditions are presented for asymptotic normality of the one-step $M$-estimators under consideration.