On the integral of the squared periodogram (Q1613586)

Let \(X_1,\dots , X_n \) be observations of a zero mean stationary Gaussian time series with spectral density \(f\), and let \(I(\lambda)\) be the periodogram \[ I(\lambda)=(2\pi n)^{-1}\left |\sum _{t=1}^n X_t e^{-it\lambda }\right |^2 . \] For a function \(\phi \) consider the functional \[ g_n(I) = \int _{-\pi }^{\pi } \eta (\lambda)\phi (I(\lambda))d\lambda , \] where \(\eta \) is a continuous even function, and its discrete version \[ b_n(I)=2\pi n^{-1}\sum _{i=1}^{n-1} \eta (\lambda _i) \phi (I(\lambda _i)), \] where \(\lambda _i=2\pi in^{-1}, \;i = 1,\dots , {n-1}.\) Here, it is proved that if \(\{X_t\}\) is Gaussian white noise and \(\phi \) a non-linear function (especially, \(\phi (\lambda) = \lambda ^2 \) is considered), then the asymptotic variance of \(g_n(I)\) is not the same as that of \(b_n(I)\). For a general stationary Gaussian sequence, the joint asymptotic distribution of the vector \[ \left (\int _{-\pi } ^\pi I_n(\lambda)d\lambda , \int _ {-\pi }^\pi I_n^2(\lambda) d\lambda \right) \] is established, where \(I_n(\lambda)\) is the mean corrected periodogram. The result is then extended to the asymptotic distribution of a goodness-of-fit test statistic for testing hypotheses on spectral densities.