Geometry of Gaussian signals

TL;DR

This paper analyzes the statistical properties of Gaussian signals with Fourier modes decaying as 1/q^α, revealing universal behavior of moments in small windows and boundary condition dependence for larger windows.

Contribution

It computes moments of Gaussian signals' width distribution, showing universality in small windows and boundary condition effects in larger windows.

Findings

Moments become universal as window size approaches zero.

Distribution independent of α for α > 3 in small-window limit.

Boundary conditions influence properties at larger window sizes.

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 $\alpha$.