A NOTE ON WEAK-CONVERGENCE OF DENSITY ESTIMATORS IN HILBERT-SPACES

In this note the impossibility of weak convergence of the density esti mator in L-2(R) is established. On the other hand a class of non-trivi al transforms of the estimator is specified that do yield a Gaussian l imiting measure on L-2(R). As a corollary we obtain a result for weak convergence of certain quadratic functionals that in a sense complemen ts a result by Bickel and Rosenblatt (1973).