Geometry of Gaussian signals

Details Geometry of Gaussian signals Geometry of Gaussian signals

2005-03-06

arxiv.org/abs/cond-mat/0503134v1

J. Stat. Mech. (2005) L08001

Physics

Condensed Matter

Disordered Systems and Neural Networks

4 pages, 3 figures

Scientific paper

10.1088/1742-5468/2005/08/L08001

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 $\alpha$.

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