Higher moments of weighted integrals of non-Gaussian fields

In general, the exact probability distribution of a definite integral of a given non-Gaussian random field is not known. Some information about this unknown distribution can be obtained from the 3rd and 4th moment of the integral. Approximations to these moments can be calculated by discretizing the integral and replacing the integrand by third-degree polynomials of correlated Gaussian variables which reproduce the first four moments and the correlation function of the field correctly. The method described (see Ditlevsen O, Mohr G, Hoffmeyer P. Integration of non-Gaussian fields. Probabilistic engineering mechanics, 1996) based on these ideas is discussed and further developed and used in a computer program which produces fairly accurate approximations to the mentioned moments with no restrictions put on the weight function applied to the field and the correlation function of the field. A pathological example demonstrating the limitations of the method is given.