= Exponential limit of a one-sided periodogram
{title2=$I_T(\omega)/s(\omega)\xrightarrow{d}\operatorname{Exp}(1)$}
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>.
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