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Asymptotics for Lp-norms of Fourier series density estimators

Authors:	Lajos Horváth

Institution:	1. Bolyai Institute, Szeged University, Aradi vértanúk tere 1, H-6720, Szeged, Hungary 2. Department of Mathematics, University of Utah, 84112, Salt Lake City, Utah, USA

Abstract:	LetX ₁, X₂, , be a sequence of independent, identically distributed bounded random variables with a smooth density functionf. We prove that $\int_a^b \| f_{m,n} (t) - f(t)\|^p w(t)dt(1 \leqslant p< \infty )$ is asymptotically normal, wheref _{m, n} is the Fourier series density estimator offandw is a nonnegative weight function.Communicated by Edward B. Saff.AMS classification: Primary 60F05, 60F25; Secondary 62G05.

Keywords:	and phrases" target="_blank"> and phrases Fourier series Wiener process Stationary Gaussian processes Central limit theorem Density estimate
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