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Weak dependence. With examples and applications. (English) Zbl 1165.62001

Lecture Notes in Statistics 190. New York, NY: Springer (ISBN 978-0-387-69951-6/pbk). xiv, 318 p. (2007).
While for many years researchers were mostly working with independent samples, recent developments of methods for dependent data have moved the interest to such more complicated structures. There are numerous examples where dependent data occur. This leads to the problem of developing asymptotic results and limit theorems for sequences of dependent random variables.
P. Doukhan, and S. Louhichi [“A new weak dependence condition and applications to moment inequalities.” Stochastic Processes Appl. 84, No. 2, 313–342 (1999; Zbl 0996.60020)] proposed a new concept of weak dependence which is more general than mixing and is suitable for almost all classes of processes of interest in statistics. Weak dependence refers to how a stochastic relationship between random variables decreases as the separation between them increases. The idea relates to measure asymptotic independence of a random process. This book continues this idea and provides a detailed description of the notion of weak dependence as well as properties and applications.
As the authors say in the Preface, the book is organized in four parts: definitions and models, tools, limit theorems and applications. A brief introduction to the concept can be found in Chapter 1. Different classes of weakly dependent sequences are described in Chapter 2. Chapter 3 contains examples of random sequences with weak dependence, including Bernoulli shifts, Markov sequences and several time series processes. Chapters 4 and 5 are devoted to develop theory to base the forthcoming results in the later chapters and especially inequalities and moment bounds.
Chapters 6 through 9 are devoted to developing limit theorems and asymptotic results, like strong laws of large numbers (Chapter 6), central limit theorems (Chapter 7), functional central limit theorems (Chapter 8), and laws of iterated logarithms (Chapter 9). Chapters 10 to 13 present interesting applications like empirical processes, functional and spectral estimation, and econometric applications respectively. The applications also refer to practical statistical problems like nonparametric statistics, spectral analysis, econometrics and resampling, offering robustness against deviations from standard independence assumptions. The book ends up with a detailed bibliography including some early attempts to the topic.
Overall the book is neatly written, however, it is quite dense and hard for researchers without strong background on the topic. The exposition is too technical in several points. Despite its complexity notations are very consistent and this helps the readers considerably. On the other hand, the book is very rich in its material as it contains earlier works on dependence and tries to mention 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 like stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory which provide good insight.

MSC:

62-02 Research exposition (monographs, survey articles) pertaining to statistics
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60F05 Central limit and other weak theorems
60G99 Stochastic processes
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
62H20 Measures of association (correlation, canonical correlation, etc.)

Citations:

Zbl 0996.60020
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