On the weak convergence of kernel density estimators in Lp spaces

Gilles Stupfler

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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.

Keywords: Economie; quantitative (search for similar items in EconPapers)
Date: 2014-10
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Published in Journal of Nonparametric Statistics, 2014, 26 (4), pp.721--735. ⟨10.1080/10485252.2014.949707⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01474248

DOI: 10.1080/10485252.2014.949707

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