Geometry of Gaussian signals

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.

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