7. Non-linear Processes

This chapter aims at describing stationary sequences generated from independent identically distributed samples \((\xi _n)_{n\in \mathbb {Z}}\). Most of the material in this chapter is specific to this monograph so that we do not provide a global reference. However Rosenblatt (Stationary processes and random fields. Birkhäuser, Boston, 1985) performs an excellent approach to modelling. Generalized linear models are presented in Kedem and Fokianos (Regression models for time series analysis. Wiley, Hoboken, 2002). The Markov case has drawn much attention, see Duflo (Random iterative models. Springer, New-York, 1996), and for example Douc et al. (Nonlinear time series: theory, methods, and applications with R examples. CRC Press, Chapman & Hall, Boca Raton, 2015) for the estimation of such Markov models. Many statistical models will be proved in this way. The organization follows the order from natural extensions of linearity to more general settings. From linear processes it is natural to build polynomial models or their limits. Then we consider more general Bernoulli shift models to define recurrence equations besides the standard Markov setting.