The Multiple Wiener-Itô Integral

A basic tool of our investigations of nonlinear problems of time series analysis is the multiple Wiener-Itô integral. In the time domain Wiener started to investigate the stationary functionals of the Brownian motion processes in terms of higher order stochastic integrals. He developed the so called chaotic series representations [146]. The frequency domain analysis of stationary flows in the space of £2 functionals of standard Wiener processes came into the picture when Rosenblatt’s noncentral limit problem was studied in more general circumstances by Sinai [115], Dobrushin[33], Dobrushin and Major [34]. See in Major [78] for details. We give here some alternatives for the definition of multiple Wiener-Itô integral for a better understanding of this technique. The chaotic representation of a stationary subordinated processes will be considered as the generalization of the classic spectral representation of stationary random processes. Two particular cases are studied in detail. One is the closest possible process to the Gaussian stationary processes which has only first, i.e., Gaussian and second order multiple terms. The second is an appropriate function of a Gaussian stationary process.