On the weak convergence of kernel density estimators in Lp spaces

Since its introduction, the pointwise asymptotic properties of the kernel estimator of a probability density function in finite dimension, as well as the asymptotic behaviour of its integrated errors, have been studied in great detail. Its weak convergence in functional spaces, however, is a more difficult problem. In this paper, we show that any Borel measurable weak limit of the (properly centred and rescaled) kernel density estimator must be 0. We also provide simple conditions for proving or disproving the existence of this Borel measurable weak limit