A Note on Weak Convergence of Density Estimators in Hilbert Spaces

In this note the impossibility of weak convergence of the density estimator in is established. On the other hand a class of non-trivial transforms of the estimator is specified that do yield a Gaussian limiting measure on . As a corollary we obtain a result for weak convergence of certain quadratic functionals that in a sense complements a result by Bickel and Rosenblatt (1973).