= Finite-record Fourier covariance bound
{title2=$|\operatorname{Cov}(A_T,B_T)|\leq\frac2{\pi T}\sum_{k<T}k|\gamma_k|$}
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.
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