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Presenting tools to aid understanding of asymptotic theory and weakly dependent processes, this book is devoted to inequalities and limit theorems for sequences of random variables that are strongly mixing in the sense of Rosenblatt, or absolutely regular.
The first chapter introduces covariance inequalities under strong mixing or absolute regularity. These covariance inequalities are applied in Chapters 2, 3 and 4 to moment inequalities, rates of convergence in the strong law, and central limit theorems. Chapter 5 concerns coupling. In Chapter 6 new deviation inequalities and new moment inequalities for partial sums via the coupling lemmas of Chapter 5 are derived and applied to the bounded law of the iterated logarithm. Chapters 7 and 8 deal with the theory of empirical processes under weak dependence. Lastly, Chapter 9 describes links between ergodicity, return times and rates of mixing in the case of irreducible Markov chains. Each chapter ends with a set of exercises.
The book is an updated and extended translation of the French edition entitled "Théorie asymptotique des processus aléatoires faiblement dépendants" (Springer, 2000). It will be useful for students and researchers in mathematical statistics, econometrics, probability theory and dynamical systems who are interested in weakly dependent processes.
List of contents
Introduction.- Variance of partial sums.- Algebraic moments. Elementary exponential inequalities.- Maximal inequalities and strong laws.- Central limit theorems.- Coupling and mixing.- Fuk-Nagaev inequalities, applications.- Empirical distribution functions.- Empirical processes indexed by classes of functions.- Irreducible Markov chains.- Appendices.- References.- Index.
About the author
Emmanuel Rio, started his career in 1987 as a mathematics assistant in Paris-Sud University. From 1990 to 2000 he was a CNRS researcher in the Probability and Statistics team of Paris-Sud University. Since 2000 he is professor in the department of mathematics of the University of Versailles Saint-Quentin en Yvelines.
Summary
Presenting tools to aid understanding of asymptotic theory and weakly dependent processes, this book is devoted to inequalities and limit theorems for sequences of random variables that are strongly mixing in the sense of Rosenblatt, or absolutely regular.
The first chapter introduces covariance inequalities under strong mixing or absolute regularity. These covariance inequalities are applied in Chapters 2, 3 and 4 to moment inequalities, rates of convergence in the strong law, and central limit theorems. Chapter 5 concerns coupling. In Chapter 6 new deviation inequalities and new moment inequalities for partial sums via the coupling lemmas of Chapter 5 are derived and applied to the bounded law of the iterated logarithm. Chapters 7 and 8 deal with the theory of empirical processes under weak dependence. Lastly, Chapter 9 describes links between ergodicity, return times and rates of mixing in the case of irreducible Markov chains. Each chapter ends with a set of exercises.The book is an updated and extended translation of the French edition entitled "Théorie asymptotique des processus aléatoires faiblement dépendants" (Springer, 2000). It will be useful for students and researchers in mathematical statistics, econometrics, probability theory and dynamical systems who are interested in weakly dependent processes.
