On the weak convergence of the kernel density estimator in the uniform topology

The pointwise asymptotic properties of the Parzen-Rosenblatt kernel estimator fˆn of a probability density function f on Rd have received great attention, and so have its integrated or uniform errors. It has been pointed out in a couple of recent works that the weak convergence of its centered and rescaled versions in a weighted Lebesgue Lp space, 1≤p\textless∞, considered to be a difficult problem, is in fact essentially uninteresting in the sense that the only possible Borel measurable weak limit is 0 under very mild conditions. This paper examines the weak convergence of such processes in the uniform topology. Specifically, we show that if fn(x)=E(fˆn(x)) and (rn) is any nonrandom sequence of positive real numbers such that rn/n√→0 then, with probability 1, the sample paths of any tight Borel measurable weak limit in an ℓ∞ space on Rd of the process rn(fˆn−fn) must be almost everywhere zero. The particular case when the estimator fˆn has continuous sample paths is then considered and simple conditions making it possible to examine the actual existence of a weak limit in this framework are provided.