If I have a discrete time series $x(t_i)$, and each of the $x(t_{i})$ are normally distributed, i.e., come from a Gaussian distribution with mean zero and variance one, would a windowed finite Fourier transform of $x(t_0)$ through $x(t_{N-1})$ also be Gaussian distributed? In other words, would the real and imaginary parts of:
$y(f) = \sum_{t=0}^{N-1}exp(-i 2 \pi f t) x(t) a(t)$
also have Gaussian distributions? a(t) is a window function that decays to 0 at $t=0$ and $t=N-1$.