Abstract
We consider Gaussian signals on a scale
L, i.e. random
functions u(t)
(
) with independent Gaussian Fourier modes of variance
∼1/qα, and compute their statistical properties in small windows 
. We determine moments of the probability distribution of the mean square width of
u(t) in powers of
the window size δ. These moments become universal in the small-window limit 
, but depend strongly on the boundary conditions of
u(t) for
larger δ. For α>3, the probability distribution can be computed explicitly, and it is independent of
α.