Noise cascades and Levy correlations

We explore a general model of stochastic noise cascades which can be illustrated by the example of rain dropping down on the earth and then seeping through layers of ground-pouring down layer by layer. The rain represents an input noise that is assumed to be spatially uncorrelated, and each ground layer represents a stochastic convolution filter. As the input noise percolates through the layered filters spatial correlations-which are initially nonexistent-build up. We study this build-up of correlations and focus on the following question: are there universally emergent forms of spatial correlations? The answer is proved affirmative, and is shown to be uniquely characterized by power spectra that coincide with the Fourier transform of the spherically symmetric Levy distribution. We term these universally emergent spatial correlations 'Levy correlations'.