Finite-record Fourier covariance bound ($|\mathrm{Cov}(A_T,B_T)|\le \frac{2}{\pi T}{\sum }_{k<T}k|{\gamma }_k|$)

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.