Geometry of Gaussian signals

We consider Gaussian signals on a scale L, i.e. random functions u( t) (t/ L is an element of [0, 1]) with independent Gaussian Fourier modes of variance similar to 1/q(alpha), and compute their statistical properties in small windows t/ L is an element of [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. These moments become universal in the small-window limit delta << 1, but depend strongly on the boundary conditions of u( t) for larger delta. For alpha > 3, the probability distribution can be computed explicitly, and it is independent of alpha.