CUMULANT SERIES EXPANSION OF HYBRID NONLINEAR MOMENTS OF N-VARIATES

In this correspondence, the extension to functions or several variables of a theorem for nonlinear transformations of random variables, recently appearing in [1], is presented. It is shown that, under general conditions, the n-fold moment between a random variable X1 and a nonlinearity of other n - 1 variables (X2, ..., X(n)) can be represented by a series of higher order cumulants. The coefficients of this cumulant expansion are the expected values of the partial derivatives of the function describing the nonlinearity. The worthwhile case of fourfold moments involving nonlinearities of Gaussian variates is directly derived by retaining those terms of the expansion corresponding to first- and second-order nonzero cumulants.