Bunches of random cross-correlated sequences

Abstract

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 correlation matrix 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 the method 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 Lévy 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.