The periodogram of an i.i.d. sequence.
type: Item
instance of: scholarly article
title: The periodogram of an i.i.d. sequence.
zbMATH Open document ID: 1046.62097
DOI: 10.1016/S0304-4149(00)00077-6
published in: Stochastic Processes and their Applications
publication date: 2004-09-22T00:00:00Z
review text: Let \(\{Z_ t\}\) be an i.i.d. sequence such that \(EZ_ t=0\) and \(EZ_ t^ 2 =1\). Define \[ d_ n(x)=(2\pi n)^ {-1/2}\sum _ {t=1}^ n Z_ t e^ {itx} \quad\text{and}\quad \widetilde n=[(n-1)/2]. \] The periodogram is \(I_ n(x)=| d_ n(x)| ^ 2\) and the Fourier frequences are \(x_ k=2\pi k/n\) for \(k=1,\dots ,\widetilde n\). Consider a function \(\phi \) and real numbers \(\beta _ {n,k}\) such that \(\sum _ {k=1} ^ {\widetilde n} \beta _ {n,k}^ 2 =1\). If \(\{Z_ t\}\) is Gaussian then under some general conditions the random variable \(S_ n(\phi )=\sum _ {k=1} ^ {\widetilde n}\beta _ {n,k} \phi [2\pi I_ n(x_ k)]\) is asymptotically normal. The authors present theory for asymptotics of weighted sums of nonlinear functionals of the periodogram of an i.i.d. sequence. They show that the sums are asymptotically normal under conditions close to those which are sufficient in the Gaussian case. The theory includes also a functional central limit theorem for the empirical spectral distribution function \[ \widehat F_ n(x) = {\widetilde n}^ {-1} \sum _ {k=1} ^ {\widetilde n}{\mathbf 1}_ {[0,x]} [2\pi I_ n(x{_ k})], \quad x\geq 0. \] The main tool in deriving the results are Edgeworth expansions for moments of smooth functions.
Mathematics Subject Classification ID: 62M15
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 2102387
zbMATH Keywords: periodogram
zbMATH Keywords: functional central limit theorem
zbMATH Keywords: empirical spectral distribution function
zbMATH Keywords: Edgeworth expansions
author: Gilles Faÿ
author: Philippe Soulier
reviewed by: Jiří Anděl
MaRDI profile type: Publication
cites work: Q4097679
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description: scientific article; zbMATH DE number 2102387