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.
