Paul Doukhan – författare

The purpose of this book is to give a detailed account of some recent devel- ments in the ?eld of probability and statistics for dependent data. It covers a wide range of topics from Markov chains theory, weak dependence, dynamical system to strong dependence and their applications. The title of this book has been somehow borrowed from the book ”Dependence in Probability and Statistics: a Survey of Recent Result” edited by Ernst Eberlein and Murad S. Taqqu, Birkh¨ auser (1986), which could serve as an excellent prerequisite for reading this book. We hope that the reader will ?nd it as useful and stimulating as the previous one. This book was planned during a conference, entitled “STATDEP2005: Statistics for dependent data”, organized by the Statistical Laboratory of the CREST (Research Center in Economy and Statistics), in Paris/Malako?, under the auspices of the French State Statistical Institute, INSEE. See http://www.crest.fr/pageperso/statdep2005/home.htm for some r- rospective informations. However this book is not a conference proceeding. This conference has witnessed the rapid growth of contributions on dep- dent data in the probabilistic and statistical literature and the need for a book covering recent developments scattered in various probability and s- tistical journals. To achieve such a goal, we have solicited some participants of the conferences as well as other specialists of the ?eld.

The purpose of this book is to give a detailed account of some recent devel- ments in the ?eld of probability and statistics for dependent data. It covers a wide range of topics from Markov chains theory, weak dependence, dynamical system to strong dependence and their applications. The title of this book has been somehow borrowed from the book ”Dependence in Probability and Statistics: a Survey of Recent Result” edited by Ernst Eberlein and Murad S. Taqqu, Birkh¨ auser (1986), which could serve as an excellent prerequisite for reading this book. We hope that the reader will ?nd it as useful and stimulating as the previous one. This book was planned during a conference, entitled “STATDEP2005: Statistics for dependent data”, organized by the Statistical Laboratory of the CREST (Research Center in Economy and Statistics), in Paris/Malako?, under the auspices of the French State Statistical Institute, INSEE. See http://www.crest.fr/pageperso/statdep2005/home.htm for some r- rospective informations. However this book is not a conference proceeding. This conference has witnessed the rapid growth of contributions on dep- dent data in the probabilistic and statistical literature and the need for a book covering recent developments scattered in various probability and s- tistical journals. To achieve such a goal, we have solicited some participants of the conferences as well as other specialists of the ?eld.

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.

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.

Mixing is concerned with the analysis of dependence between sigma-fields defined on the same underlying probability space. It provides an important tool of analysis for random fields, Markov processes, central limit theorems as well as being a topic of current research interest in its own right. The aim of this monograph is to provide a study of applications of dependence in probability and statistics. It is divided in two parts, the first covering the definitions and probabilistic properties of mixing theory. The second part describes mixing properties of classical processes and random fields as well as providing a detailed study of linear and Gaussian fields. Consequently, this book will provide statisticians dealing with problems involving weak dependence properties with a powerful tool.

Time series research has been an area of considerable research activity over the past several decades. The essential ingredient --- the notion of time-dependence --- is required for measuring and then accurately predicting data to construct suitable models for diverse phenomena. This fairly self-contained volume, written by leading experts in their respective fields, especially focuses on the theoretical concepts, methodologies, and practical applications pertaining to self-similar processes and long-range dependent phenomena. Graduate students, researchers, and professionals in industry will benefit from the book.

Mixing is concerned with the analysis of dependence between sigma-fields defined on the same underlying probability space. It provides an important tool of analysis for random fields, Markov processes, central limit theorems as well as being a topic of current research interest in its own right. The aim of this monograph is to provide a study of applications of dependence in probability and statistics. It is divided in two parts, the first covering the definitions and probabilistic properties of mixing theory. The second part describes mixing properties of classical processes and random fields as well as providing a detailed study of linear and Gaussian fields. Consequently, this book will provide statisticians dealing with problems involving weak dependence properties with a powerful tool.

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.

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.

 

This volume contains several contributions on the general theme of dependence for several classes of stochastic processes, andits implicationson asymptoticproperties of various statistics and on statistical inference issues in statistics and econometrics. The chapter by Berkes, Horvath and Schauer is a survey on their recent results on bootstrap and permutation statistics when the negligibility condition of classical central limit theory is not satis ed. These results are of interest for describing the asymptotic properties of bootstrap and permutation statistics in case of in nite va- ances, and for applications to statistical inference, e.g., the change-point problem. The paper by Stoev reviews some recent results by the author on ergodicity of max-stable processes. Max-stable processes play a central role in the modeling of extreme value phenomena and appear as limits of component-wise maxima. At the presenttime,arathercompleteandinterestingpictureofthedependencestructureof max-stable processes has emerged,involvingspectral functions, extremalstochastic integrals, mixed moving maxima, and other analytic and probabilistic tools.For statistical applications, the problem of ergodicity or non-ergodicity is of primary importance.