Electronic Communications in Probability

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

Gilles Stupfler

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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.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 17, 13 pp.

Dates
Received: 16 October 2015
Accepted: 12 February 2016
First available in Project Euclid: 25 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1456412897

Digital Object Identifier
doi:10.1214/16-ECP4638

Mathematical Reviews number (MathSciNet)
MR3485386

Zentralblatt MATH identifier
1338.60100

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 62G07: Density estimation 62G20: Asymptotic properties

Keywords
kernel density estimator weak convergence $\ell ^{\infty }$ space tightness

Rights
Creative Commons Attribution 4.0 International License.

Citation

Stupfler, Gilles. On the weak convergence of the kernel density estimator in the uniform topology. Electron. Commun. Probab. 21 (2016), paper no. 17, 13 pp. doi:10.1214/16-ECP4638. https://projecteuclid.org/euclid.ecp/1456412897


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