Exponential limit of a one-sided periodogram ($I_T^{+}(\omega )/s(\omega )\stackrel{d}{\to }\mathrm{Exp}(1)$)

Let $X_t$ be a centered real stationary Gaussian time series with absolutely summable autocovariances and two-sided angular-frequency spectral density $f$. The parent periodogram uses $I_T(\omega )=(2\pi T{)}^{-1}|{\sum }_{t=1}^T{X}_t{e}^{-it\omega }{|}^2$. At an interior positive frequency define the one-sided quantitiesThus the doubling applies to both the statistic and its target spectral density.

For Fourier-grid frequencies ${\omega }_T=2\pi k_T/T\to \omega \in (0,\pi )$ with $f(\omega )>0$, setThen $I_T^{+}({\omega }_T)=A_T^2+B_T^2$. Summability makes their variances tend to $f(\omega )=s(\omega )/2$ and their covariance tend to zero. Joint Gaussianity therefore givesUnder these Gaussian assumptions the fourth moments also converge, so $\mathrm{var}(I_T^{+}({\omega }_T))\to s(\omega {)}^2$. An unsmoothed periodogram is therefore not a consistent estimator of a positive spectral density. At zero and the Nyquist frequency $\pi$, the sine coordinate vanishes and the real-only calibration differs; the interior doubling and two-coordinate argument do not apply there.^[ref]