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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\textless∞, the limit must be 0. We also provide simple conditions for proving or disproving the existence of this Borel measurable weak limit.
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https://hal-amu.archives-ouvertes.fr/hal-01474248
Contributor : Elisabeth Lhuillier <>
Submitted on : Wednesday, February 22, 2017 - 4:16:12 PM
Last modification on : Wednesday, August 5, 2020 - 3:18:21 AM

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Gilles Stupfler. On the weak convergence of kernel density estimators in Lp spaces. Journal of Nonparametric Statistics, 2014, 26 (4), pp.721--735. ⟨10.1080/10485252.2014.949707⟩. ⟨hal-01474248⟩

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