Windowed Fourier transform of Gaussian distributed random time series 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 $t=N-1$.

Yes, provided one assumes that the family $(x(t))_{0\leqslant t\leqslant N-1}$ is gaussian (recall that, as soon as $N\geqslant2$, this asks strictly more than each $x(t)$ being normally distributed). Then $y(f)$, the real part of $y(f)$ and the imaginary part of $y(f)$ are all **linear combinations** of a gaussian family hence they are normally distributed (complex-valued for $y(f)$ and real-valued for the others).

**Edit:** A standard way to ensure that a family is gaussian is to assume that each random variable is normally distributed and that the family is independent.