Geometry of Gaussian signals

We consider Gaussian signals, i.e. random functions $u(t)$ ($t/L \in [0,1]$) with independent Gaussian Fourier modes of variance $\sim 1/q^α$, 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 $δ\ll 1$, become universal, whereas they strongly depend on the boundary conditions of $u(t)$ for larger $δ$. For $α> 3$, the probability distribution is computed in the small-window limit and shown to be independent of $α$.

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