Ebook: Mixing: Properties and Examples

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.

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These notes are devoted to the study of mixing theory. The underlying goal is to provide statisticians dealing with problems involving weak dependence properties, with a powerful and easy tool. Up to now, this approach to dependence has been mainly considered from an abstract point of view. For an excellent review on this subject we refer to : "Dependence in Probability and Statistics, a Survey of Recent Results" ('). The aim of this work is to study applications of these results. We obtain bounds for the decay of mixing coefficient sequences associated to random processes or to random fields which are actually used in Statistics. In some cases we will give counterexamples which show that some frequently held ideas are wrong. In fact, it would be of little interest to study a probabilistic technique with no field of application.