## Electronic Communications in Probability

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

Gilles Stupfler

#### 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
Accepted: 12 February 2016
First available in Project Euclid: 25 February 2016

https://projecteuclid.org/euclid.ecp/1456412897

Digital Object Identifier
doi:10.1214/16-ECP4638

Mathematical Reviews number (MathSciNet)
MR3485386

Zentralblatt MATH identifier
1338.60100

#### 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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