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Asymptotics for \(L_ p\)-norms of Fourier series density estimators. (English) Zbl 0706.62042

Let \(X_ 1,X_ 2,..\). be a sequence of independent, identically distributed bounded random variables with a smooth density function f and \(f_{m,n}\) be the Fourier series density estimator of f. Consider the \(L_ p\) \((1\leq p<\infty)\) distance between \(f_{m,n}\) and f \[ \int^{b}_{a}| f_{m,n}(t)-f(t)|^ p w(t)dt, \] where w is a nonnegative weight function. For \(p=2\), some central limit theorems have been obtained by some authors. In this paper, a theorem, which is an \(L_ p\)-version of \(L_ 2\)-results, is proved by a different method, which is used in the proof of asymptotic normality of the \(L_ p\)-norm of kernel estimators.
Reviewer: Zengyan Lin

MSC:

62G05 Nonparametric estimation
60F05 Central limit and other weak theorems
60F25 \(L^p\)-limit theorems
62G20 Asymptotic properties of nonparametric inference
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