ONE METHOD OF CONSTRUCTING NON-GAUSSIAN RANDOM FUNCTIONS WITH GIVEN PDF AND GIVEN SPECTRUM

摘要

We meet non-Gaussian random functions in many physical applications. The turbulence theory, nonlinear random waves, and nonlinear circuits are some of examples. Gaussian random functions f (R), where R is a point in n-dimensional real space, are completely determined by the first moment < f (R_n) > and the second moment < f(R_1) f (R_2) > . Non-Gaussian random functions require the infinite number of moments (or cumulants) < f (R_1) f (R_2) • • • f (R_m) >, m = 1,2, ...,∞. For the Gaussian case, for any order N we know the joint probability density function. For non-Gaussian case, there exist not too many examples of joint PDF of an arbitrary order N. The common method of describing of non-Gaussian distributions is based on cumulant expansions. It is known, however that any truncated cumulant expansion leads to false negative probabilities. We develop the method of description of multivariate non-Gaussian distributions based on approximation of an arbitrary multivariate PDF by superposition of shifted Gaussian multivariate distributions having different correlation functions. This method is free from negative probabilities and it allows to satisfy some additional conditions. We present an example of joint PDF of a random surface with the given spatial spectrum and given joint PDF of two principal slopes.