ENSAE, Laboratoire de Statistique du CREST (Paris Tech), France
SAMOS-MATISSE (Statistique Appliquée et Modélisation Stochastique), Centre d’Economie de la Sorbonne Université Paris 1-Panthéon-Sorbonne CNRS, France
Time series and random ?elds are main topics in modern statistical techniques. They are essential for applications where randomness plays an important role. Indeed, physical constraints mean that serious modelling cannot be done - ing only independent sequences. This is a real problem because asymptotic properties are not always known in this case. Thepresentworkisdevotedtoprovidingaframeworkforthecommonlyused time series. In order to validate the main statistics, one needs rigorous limit theorems. In the ?eld of probability theory, asymptotic behavior of sums may or may not be analogous to those of independent sequences. We are involved with this ?rst case in this book. Very sharp results have been proved for mixing processes, a notion int- duced by Murray Rosenblatt [166]. Extensive discussions of this topic may be found in his Dependence in Probability and Statistics (a monograph published by Birkhau ¨ser in 1986) and in Doukhan (1994) [61], and the sharpest results may be found in Rio (2000)[161]. However, a counterexample of a really simple non-mixing process was exhibited by Andrews (1984) [2]. The notion of weak dependence discussed here takes real account of the available models, which are discussed extensively. Our idea is that robustness of the limit theorems with respect to the model should be taken into account. In real applications, nobody may assert, for example, the existence of a density for the inputs in a certain model, while such assumptions are always needed when dealing with mixing concepts.
Reviews
From the reviews:
"I appreciate this book as a very welcome and thorough discussion of the actual state-of-the art in the modeling of dependence structures. It provides a large number of motivating examples and applications, rigorous proofs, and valuable intuitions for the willing and mathematically well-trained reader with essential prior knowledge of the mathematical prerequisites of weak dependence … . It is … the book to those researchers already aware of the necessity of the methods discussed here." (Harry Haupt, Advances in Statistical Analysis, Vol. 93, 2009)
"This book … provides a detailed description of the notion of weak dependence as well as properties and applications. … Overall the book is neatly written … . the book is very rich in its material as it contains earlier works on dependence and … show a lot of applications of the theory. It also contains a large number of examples and expositions of the idea of weak dependence in models … which provide good insight." (Dimitris Karlis, Zentralblatt MATH, Vol. 1165, 2009)
Authors and Affiliations
Laboratoire de Statistique, Théorique et Appliquée, Université Paris 6, 75013 Paris, France
Jérôme Dedecker
ENSAE, Laboratoire de Statistique du CREST (Paris Tech), France
Paul Doukhan
SAMOS-MATISSE (Statistique Appliquée et Modélisation Stochastique), Centre d’Economie de la Sorbonne Université Paris 1-Panthéon-Sorbonne CNRS, France
Paul Doukhan
Equipe MORSE UMR MIA518 INRA, F-75005 Paris, France
Gabriel Lang
Escuela de Matematica, Universidad Central de Venezuela, Caracas 1041-A, Venezuela
León R. José Rafael
Laboratoire de Mathématiques, Université Paris-Sud–Bât 425, France
Sana Louhichi
Génie Mathématique et Modélisation, Laboratoire de Statistique et Probabilités, INSA Toulouse, France
Clémentine Prieur
Bibliographic Information
Book Title: Weak Dependence: With Examples and Applications
Authors: Jérôme Dedecker, Paul Doukhan, Gabriel Lang, León R. José Rafael, Sana Louhichi, Clémentine Prieur
