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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 L'evy distribution. We term these universally emergent spatial correlations 'L'evy correlations'. [ABSTRACT FROM AUTHOR]

The statistical properties of random cross-correlated sequences constructed by the convolution method (likewise referred to as the Rice or the inverse Fourier transformation) are examined. We clarify the meaning of the filtering function--the kernel of the convolution operator--and show that it is the value of the cross-correlation function which describes correlations between the initial white noise and constructed correlated sequences. The matrix generalization of this method for constructing a bunch of N cross-correlated sequences is presented. Algorithms for their generation are reduced to solving the problem of decomposition of the Fourier transform of the correlationmatrix into a product of two mutually conjugate matrices. Different decompositions are considered. The limits of weak and strong correlations for the one-point probability and pair correlation functions of sequences generated by themethod under consideration are studied. Special cases of heavy-tailed distributions of the generated sequences are analyzed. We show that, if the filtering function is rather smooth, the distribution function of generated variables has the Gaussian or L'evy form depending on the analytical properties of the distribution (or characteristic) functions of the initial white noise. Anisotropic properties of statistically homogeneous random sequences related to the asymmetry of a filtering function are revealed and studied. These asymmetry properties are expressed in terms of the third- or fourth-order correlation functions. Several examples of the construction of correlated chains with a predefined correlation matrix are given. [ABSTRACT FROM AUTHOR]

We analyze the signum-generation method for creating random dichotomic sequences with prescribed correlation properties. The method is based on a binary mapping of the convolution of continuous random numbers with some function originated from the Fourier transform of a binary correlator. The goal of our study is to reveal conditions under which one can construct binary sequences with a given pair correlator. Our results can be used in the construction of superlattices and waveguides with selective transport properties. [ABSTRACT FROM AUTHOR]