Periodogram ($I_T(\lambda )$)

For a centered record of length $T$, the periodogram is the squared modulus of its discrete Fourier transform, with normalizationUnder suitable short-memory assumptions its expectation approaches the spectral density of a stationary process, but its variance generally does not vanish. Smoothing nearby frequencies can provide an estimate with statistical consistency.

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]

For a centered stationary real Gaussian sequence, let $A_T$ and $B_T$ be its cosine and sine coefficients with normalization $1/\sqrt{\pi T}$ at a nonzero Fourier frequency. Grouping the double covariance sum by lag and extending each lag sum to a whole record leaves at most $2k$ omitted bounded products. The full cosine-sine sum is zero, giving the bound. If $\sum k|{\gamma }_k|<\infty$, the two coefficients become asymptotically uncorrelated, and their limiting joint normal distribution consequently has independent components.

Average neighboring periodogram ordinates with nonnegative weights summing to one. Let the number $L_T$ of averaged ordinates tend to infinity while their frequency span, of order $L_T/T$, tends to zero. For a smooth spectral density and suitable short-memory or fourth-order dependence conditions, the bias tends to zero and variance is of order $1/L_T$. Smooth second-order spectral density alone does not ensure this: a time-constant random scale multiplying independent white noise has a smooth unconditional spectrum but a nonergodic random limiting local power.