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

At an interior frequency, the real and imaginary Fourier coefficients of a short-memory Gaussian time series converge jointly to independent centered normal variables, each with variance $s(\omega )/2$, where $s$ is the one-sided spectral density. Their squared sum therefore converges to $s(\omega ){\chi }_2^2/2$. Its variance tends to $s(\omega {)}^2$, so an unsmoothed periodogram is not consistent when $s(\omega )>0$. Averaging nearby ordinates can reduce this persistent variance.