  We consider Gaussian signals, i.e. random functions $u(t)$ ($t/L \in [0,1]$)
with independent Gaussian Fourier modes of variance $\sim 1/q^{\alpha}$, and
compute their statistical properties in small windows $[x, x+\delta]$. We
determine moments of the probability distribution of the mean square width of
$u(t)$ in powers of the window size $\delta$. We show that the moments, in the
small-window limit $\delta \ll 1$, become universal, whereas they strongly
depend on the boundary conditions of $u(t)$ for larger $\delta$. For $\alpha >
3$, the probability distribution is computed in the small-window limit and
shown to be independent of $\alpha$.
