On the weak convergence of kernel density estimators in L^p spaces

Abstract

Since its introduction, the pointwise asymptotic properties of the kernel estimator fˆ_n of a probability density function f on ℝ^d, as well as the asymptotic behaviour of its integrated errors, have been studied in great detail. Its weak convergence in functional spaces, however, is a more difficult problem. In this paper, we show that if f_n(x)=(fˆ_n(x)) and (r_n) is any nonrandom sequence of positive real numbers such that r_n/√n→0 then if r_n(fˆ_n−f_n) converges to a Borel measurable weak limit in a weighted L^p space on ℝ^d, with 1≤p<∞, the limit must be 0. We also provide simple conditions for proving or disproving the existence of this Borel measurable weak limit.

Keywords:

• kernel density estimator
• weak convergence
• L^p space

AMS Subject Classification:

• 62G07
• 62G20
• 60F17

Acknowledgements

I would like to thank two anonymous referees for their useful suggestions and comments.