Mathematical Methods for Describing the non-Gaussian Random Variables and Processes

Abstract

Mathematical methods for describing univariate and bivariate non-Gaussian random variables and processes in their modeling are considered. A decomposition of probability density functions (PDFs) using orthogonal polynomials is described for both univariate and bivariate PDFs. It is noted that, in both cases, decompositions using the specified orthogonal functions can be used in most cases. The area of application of the exponential PDF is shown. We give a representation of two-dimensional PDF in the form of an Edgeworth-type expansion by Hermite polynomials. The correlation between the correlation function and the bivariate PDF of a random process is shown. We analyze the representation of one-dimensional distributions by orthogonal Gramm-Charlier and Edgeworth series as well as of two-dimensional distributions in the form of a Hermite polynomial expansion. It is noted that Edgeworth’s series provides a better approximation of PDFs than the Gramm-Charlier series. It is shown that the coefficient of excess characterizes the shape of PDF. We consider the method of PDF decomposition by Laguerre polynomials which are used only for one-way PDFs. The field of Fourier series decomposition is determined. The formation and superposition methods used in the formation of random variables are described.