The periodogram of an i.i.d. sequence.
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.
empirical spectral distribution function
scientific article; zbMATH DE number 2102387