Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$.
The discrete fourier transform is defined
$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n /N}$
where $k$ belongs to the integers smaller than $N$.
I would like to know if it is true that the asymptotic probability distribution for $N \to \infty$ of the real and imaginary part of $b_k$ is the product of $2N$ gaussians (or a general gaussian, but I checked that the covariance matrix is diagonal).
Thanks in advance for any answer.