This question has been bothering me for quite some time.
Consider a complex random process in time domain $s$, whose Fourier transform is $S$. In many papers, I see that they assume:
$$ \mathbb{E}[S_p S^*_q] = \mathcal{DFT}(C); p = q, $$ where $\mathcal{DFT}$ is the Discrete Fourier Transform operator $C$ is the correlation function of the random process that is usually dependent on the time difference. $C(t_m, t_n) = C(t_m - t_n)$.
$$ \mathbb{E}[S_p S^*_q] = 0; p \neq q $$
But these are assumed if the number of samples tends to $\infty$. There is one particular paper that says that the spectral coefficients that are farthest apart from the center are usually correlated unless the spectral width (inverse of the correlation width) becomes more than 0.2 times the Nyquist interval and number of samples is more than 32.
First, I don't understand why the spectral coefficients are uncorrelated. Can there be an approximate formulation of the correlation between the spectral coefficients when the number of samples is finite?
[1] D. S. Zrnic, “Spectral Statistics for Complex Colored Discrete-Time Sequenc,” vol. 28, no. 5, pp. 5–8, 1980.