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

The pointwise asymptotic properties of the Parzen-Rosenblatt kernel estimator f ₙ of a probability density function f on R ᵈ 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<∞, 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 (f ₙ(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 ∞ space on R ᵈ of the process rₙ(f ₙ-fₙ) must be almost everywhere zero. The particular case when the estimator f ₙ 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.