Approximation and simulation of probability distributions with a variable kurtosis value

For cases when the Gaussian law does not fit a given histogram in terms of its peakedness the paper proposes a family of binormal distributions that have been constructed by joining sections of two different Gaussian laws shifted vertically or horizontally. The continuity of the approximating function itself and its derivative are provided. The obtained probability densities are unimodal and rigorously positive at any parameters which are determined in such a way that in addition to a mathematical expectancy and variance the prescribed kurtosis magnitude is provided. The adoption of the stated binormal laws in applied problems requires only recurring operations with exponential dependencies resembling Gaussian ones. Random process numerical modeling is often performed by Fourier expansion with phase shifts between harmonics chosen at random. At this, the obtained sequence has a probability density close to a normal law. For generating time histories which differ from a Gauss model in terms of the peakedness of an instantaneous-value distribution, formulae connecting the kurtosis parameter of a polyharmonic process with its amplitudes and phase angles have been derived. On this basis a computational technique for simulating pseudorandom data with controlled kurtosis due to phase variation has been developed. In this approach, amplitudes are fixed according to a frequency spectrum as in the classical procedure. Thus, the proposed method for non-Gaussian simulation does not increase power spectral error.