Stochastic Models for Time Series: (80 Mathématiques et Applications)

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.  

Table of Contents:
Part I Independence and Stationarity.- 1 Probability and Independence.- 2 Gaussian convergence and inequalities.- 3 Estimation concepts.- 4 Stationarity.- Part II Models of time series.- 5 Gaussian chaos.- 6 Linear processes.- 7 Non-linear processes.- 8 Associated processes.- Part III Dependence.- 9 Dependence.- 10 Long-range dependence.- 11 Short-range dependence.- 12 Moments and cumulants.- Appendices.- A Probability and distributions.- B Convergence and processes.- C R scripts used for the gures.- Index- List of figures.

About the Author :
Paul Doukhan is a Professor at the University of Cergy-Pontoise, Paris. He is an established researcher in the area of non-linear time series. Chiefly focusing on the dependence of stochastic processes, he has published a large number of methodological research papers and authored several books in this research area.

Review :
“We are dealing with a monograph that compiles a broad set of fundamental results on the probabilistic and statistical analysis of time series, and is mathematically rigorous and effectively suitable to support theoretical and practical re-search in the general field of stochastic processes.” (Nazare Mendes Lopes, Mathematical Reviews, July, 2019) “Although there are several books written on time series and stochastic processes, this book is the first one to present stochastic modelling approaches in linear/nonlinear time series. … This book is intended for masters and higher undergraduate students in mathematics, probability, statistics, astrophysics, biomedical engineering, and neuroscience. However, the students who want to pursue a PhD in the modelling of non-linear time series should also read this book to gain fundamental background knowledge.” (Chitaranjan Mahapatra, ISCB News, iscb.info, Issue 67, June, 2019) “The book is well-written and mathematically rigorous. The author is certainly one of the best specialists in the field worldwide. He has collected a large variety of results. To date there is no book like this. It may become the standard reference for researchers working on the topic. In summary, this is a very useful book for a researcher in probability and stochastic processes, which can also be used for under- and post-graduate courses.” (Nikolai N. Leonenko, zbMATH 1401.62007, 2019)