1 Foundations.- 1.1 Expectation of Nonlinear Functions of Gaussian Variables.- 1.2 Hermite Polynomials.- 1.3 Cumulants.- 1.4 Diagrams, and Moments and Cumulants for Gaussian Systems.- 1.5 Stationary processes and spectra.- 2 The Multiple Wiener-Itô Integral.- 2.1 Functions of Spaces $$ \overline {L_{\Phi }^{n}} $$ and $$ \widetilde{{L_{\Phi }^{n}}} $$.- 2.2 The multiple Wiener-Itô Integral of second order.- 2.3 The multiple Wiener-Itô integral of order n.- 2.4 Chaotic representation of stationary processes.- 3 Stationary Bilinear Models.- 3.1 Definition of bilinear models.- 3.2 Identification of a bilinear model with scalar states.- 3.3 Identification of bilinear processes, general case.- 3.4 Identification of multiple-bilinear models.- 3.5 State space realization.- 3.6 Some bilinear models of interest.- 3.7 Identification of GARCH(1,1) Model.- 4 Non-Gaussian Estimation.- 4.1 Estimating a parameter for non-Gaussian data.- 4.2 Consistency and asymptotic variance of the estimate.- 4.3 Asymptotic normality of the estimate.- 4.4 Asymptotic variance in the case of linear processes.- 5 Linearity Test.- 5.1 Quadratic predictor.- 5.2 The test statistics.- 5.3 Comments on computing the test statistics.- 5.4 Simulations and real data.- 6 Some Applications.- 6.1 Testing linearity.- 6.2 Bilinear fitting.- Appendix A Moments.- Appendix B Proofs for the Chapter Stationary Bilinear Models.- Appendix C Proofs for Section 3.6.1.- Appendix D Cumulants and Fourier Transforms for GARCH(1,1).- Appendix E Proofs for the Chapter Non-Gaussian Estimation.- E.0.1 Proof for Section 4.4.- Appendix F Proof for the Chapter Linearity Test.- References.
