Stochastic Processes and Long Range Dependence

Authors: Gennady Samorodnitsky



First book of its kind to offer a self-contained modern introduction to long-range dependence

Written by a leading expert in the field; the product of many years of experience and insight

Presents the reader with both the standard approach and the author’s own, less standard, approach to long-range dependence

Includes new results made by the author

This monograph is a gateway for researchers and graduate students to explore the profound, yet subtle, world of long-range dependence (also known as long memory). The text is organized around the probabilistic properties of stationary processes that are important for determining the presence or absence of long memory. The first few chapters serve as an overview of the general theory of stochastic processes which gives the reader sufficient background, language, and models for the subsequent discussion of long memory. The later chapters devoted to long memory begin with an introduction to the subject along with a brief history of its development, followed by a presentation of what is currently the best known approach, applicable to stationary processes with a finite second moment. The book concludes with a chapter devoted to the author’s own, less standard, point of view of long memory as a phase transition, and even includes some novel results.

Most of the material in the book has not previously been published in a single self-contained volume, and can be used for a one- or two-semester graduate topics course. It is complete with helpful exercises and an appendix which describes a number of notions and results belonging to the topics used frequently throughout the book, such as topological groups and an overview of the Karamata theorems on regularly varying functions.

Table of contents

Front Matter
Pages i-xi

Stationary Processes
Pages 1-26

Elements of Ergodic Theory of Stationary Processes and Strong Mixing
Pages 27-71

Infinitely Divisible Processes
Pages 73-132

Heavy Tails
Pages 133-173

Introduction to Long-Range Dependence
Pages 175-191

Second-Order Theory of Long-Range Dependence
Pages 193-228

Fractionally Differenced and Fractionally Integrated Processes
Pages 229-246

Self-Similar Processes
Pages 247-283

Long-Range Dependence as a Phase Transition
Pages 285-361

Appendix
Pages 363-404

Back Matter
Pages 405-415

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