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

Abstract

The pointwise asymptotic properties of the Parzen-Rosenblatt kernel estimator $\widehat{f} _n$ of a probability density function $f$ on $\mathbb{R} ^d$ 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 $L^p$ space, $1\leq p<\infty $, 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 $f_n(x)=\mathbb{E} (\widehat{f} _n(x))$ and $(r_n)$ is any nonrandom sequence of positive real numbers such that $r_n/\sqrt{n} \to 0$ then, with probability 1, the sample paths of any tight Borel measurable weak limit in an $\ell ^{\infty }$ space on $\mathbb{R} ^d$ of the process $r_n(\widehat{f} _n-f_n)$ must be almost everywhere zero. The particular case when the estimator $\widehat{f} _n$ 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.