On the weak convergence of kernel density estimators in Lp spaces

Abstract

Since its introduction, the pointwise asymptotic properties of the kernel estimator fˆn of a probability density function f on ℝd, 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 fn(x)=(fˆn(x)) and (rn) is any nonrandom sequence of positive real numbers such that rn/√n→0 then if rn(fˆn−fn) converges to a Borel measurable weak limit in a weighted Lp space on ℝd, 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.