An alternative regularity condition for Hájek's representation theorem (Q5903347)

Hájek's representation theorem [\textit{J. Hájek}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 14, 323-330 (1970; Zbl 0193.180)] states that under certain regularity conditions the limiting distribution of an estimator can be written as the convolution of a certain normal distribution with some other distribution. This result, originally developed for finite dimensional problems, has been extended to a number of infinite dimensional settings where it has been used, for example, to establish the asymptotic efficiency of the Kaplan-Meier estimator. The purpose of this note is to show that the somewhat unintuitive regularity condition on the estimators that is usually used can be replaced by a simple one: It is sufficient for the asymptotic information and the limiting distribution of the estimator to vary continuously with the parameter being estimated.