Mathematical Methods for Describing a non-Gaussian Probability Density Function

The analysis of mathematical methods used to describe non-Gaussian random processes is carried out. Expansion of one- and two-dimensional probability distribution functions (PDFs) using orthogonal polynomials is given. It is shown that in the named method it is possible to use an exponential PDF, in particular, Pearson's PDF. A two-dimensional PDF is presented as an expansion in terms of Hermite polynomials in an Edgeworth-type series. An expression is obtained which allows us to estimate the difference between the two-dimensional PDF of a random process and its model. The representation of one-dimensional unimodal PDFs by orthogonal series, including the Gram-Charlier and Edgeworth series, is described. It is shown that the expansion coefficients can be expressed in terms of the corresponding cumulants. The relationship of cumulants with the coefficients of skewness and kurtosis, which characterize the shape of the PDF, is presented. It is shown that for a fixed maximum order of the cumulants used, the best approximation of the PDF is provided by the Edgeworth series in comparison with the Gram-Charlier series. Expansion in Laguerre polynomials is given, which is used for one-sided PDFs defined only for positive values of the argument. In this case, the Edgeworth series corresponding to the PDF converges slowly. It is shown that the Fourier series is used in the case when the one-dimensional PDF differs from zero only on a finite interval of the argument values.