Foundations

Abstract

There are several ways to approach Hermite polynomial systems, cumulants and their relationship. Our treatment starts with a general method of calculating the expectation of nonlinear function of Gaussian random variables, see [13] and [127]. Then we define the classical Hermite polynomials and their generalization with several variables. A rather simple introduction to cumulants is given. The diagram formulae are used to show the basic connections between cumulants and Hermite polynomials. These facts will be important for the multiple Wiener-Itô stochastic integrals in the next Chapter. Some general well known theory of stationary processes as spectral representation and higher order spectra are also considered. An approximation of the spectrum and the bispectrum of some nonlinear function of a Gaussian stationary process closes the Chapter.