A Note on Weak Convergence of Density Estimators in Hilbert Spaces

Abstract

In this note the impossibility of weak convergence of the density estimator in L ^{2}({\Bbb R}) 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 L ^{2}({\Bbb R}) . 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).

AMS 1991 Subject Classifications::

• Primary: 62G05
• Secondary: 62E20

Keywords:

• Weak convergence
• density estimator
• Hilbert space
• quadratic functional