Geometry of Gaussian signals

Abstract

We consider Gaussian signals, i.e. random functions $u(t)$ ($t/L\in [0,1]$) with independent Gaussian Fourier modes of variance $\sim 1/q^{\alpha }$, and compute their statistical properties in small windows $[x,x+\delta ]$. We determine moments of the probability distribution of the mean square width of $u(t)$ in powers of the window size $\delta$. We show that the moments, in the small-window limit $\delta \ll 1$, become universal, whereas they strongly depend on the boundary conditions of $u(t)$ for larger $\delta$. For $\alpha >3$, the probability distribution is computed in the small-window limit and shown to be independent of $\alpha$.

4 pages, 3 figures

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