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- handle: 10670/1.ldtyhi
- hal: hal-01474248
- doi: 10.1080/10485252.2014.949707
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info:eu-repo/semantics/altIdentifier/doi/10.1080/10485252.2014.949707
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Gilles Stupfler, « On the weak convergence of kernel density estimators in Lp spaces », HAL-SHS : économie et finance, ID : 10.1080/10485252.2014.949707
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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.