We consider Gaussian signals, i.e. random functions u(t) (t/L ∈ [0,1]) with independent Gaussian Fourier modes of variance ∼ 1/q^α, and compute their statistical properties in small windows [x, x+δ]. We determine moments of the probability distribution of the mean square width of u(t) in powers of the window size δ. We show that the moments, in the small-window limit δ≪ 1, become universal, whereas they strongly depend on the boundary conditions of u(t) for larger δ. For α > 3, the probability

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... istribution is computed in the small-window limit and shown to be independent of α.