This book presents essential tools for modelling non-linear time series. The first part of the
book describes the main standard tools of probability and statistics that directly apply to the
time series context to obtain a wide range of modelling possibilities. Functional estimation
and bootstrap are discussed and stationarity is reviewed. The second part describes a number
of tools from Gaussian chaos and proposes a tour of linear time series models. It goes on to
address nonlinearity from polynomial or chaotic models for which explicit expansions are
available then turns to Markov and non-Markov linear models and discusses Bernoulli shifts
time series models. Finally the volume focuses on the limit theory starting with the ergodic
theorem which is seen as the first step for statistics of time series. It defines the
distributional range to obtain generic tools for limit theory under long or short-range
dependences (LRD SRD) and explains examples of LRD behaviours. More general techniques (central
limit theorems) are described under SRD mixing and weak dependence are also reviewed. In
closing it describes moment techniques together with their relations to cumulant sums as well
as an application to kernel type estimation.The appendix reviews basic probability theory facts
and discusses useful laws stemming from the Gaussian laws as well as the basic principles of
probability and is completed by R-scripts used for the figures. Richly illustrated with
examples and simulations the book is recommended for advanced master courses for
mathematicians just entering the field of time series and statisticians who want more
mathematical insights into the background of non-linear time series.
