Synopses & Reviews
This monograph is aimed at developing Doukhan/Louhichi's (1999) idea to measure asymptotic independence of a random process. The authors propose various examples of models fitting such conditions such as stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory. Most of the commonly used stationary models fit their conditions. The simplicity of the conditions is also their strength. The main existing tools for an asymptotic theory are developed under weak dependence. They apply the theory to nonparametric statistics, spectral analysis, econometrics, and resampling. The level of generality makes those techniques quite robust with respect to the model. The limit theorems are sometimes sharp and always simple to apply. The theory (with proofs) is developed and the authors propose to fix the notation for future applications. A large number of research papers deals with the present ideas; the authors as well as numerous other investigators participated actively in the development of this theory. Several applications are still needed to develop a method of analysis for (nonlinear) times series and they provide here a strong basis for such studies. Jérôme Dedecker (associate professor Paris 6), Gabriel Lang (professor at Ecole Polytechnique, ENGREF Paris), Sana Louhichi (Paris 11, associate professor at Paris 2), and Clémentine Prieur (associate professor at INSA, Toulouse) are main contributors for the development of weak dependence. José Rafael León (Polar price, correspondent of the Bernoulli society for Latino-America) is professor at University Central of Venezuela and Paul Doukhan is professor at ENSAE (SAMOS-CES Paris 1 and Cergy Pontoise) and associate editor of Stochastic Processes and their Applications. His Mixing: Properties and Examples (Springer, 1994) is a main reference for the concurrent notion of mixing.
Review
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)
Synopsis
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."
Synopsis
This book develops Doukhan/Louhichi's 1999 idea to measure asymptotic independence of a random process. The authors, who helped develop this theory, propose examples of models fitting such conditions: stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory. Applications are still needed to develop a method of analysis for nonlinear times series, and this book provides a strong basis for additional studies.
Table of Contents
Introduction.- Weak dependence.- Models.- Tools for non causal cases.- Tools for causal cases.- Applications of SLLN.- Central limit theorem.- Donsker principles.- Law of the iterated logarithm (LIL).- The empirical process.- Functional estimation.- Spectral estimation.- Econometrics and resampling.