Long range dependence and self similarity

Long-Range Dependence and Self-Similarity This modern and comprehensive guide to long-range dependence and self-similarity starts with rigorous coverage of the basics, then moves on to cover more specialized, up-to-date topics central to current research These topics concern, but are not limited to, physical models that give rise to long-range dependence and self-similarity; central and non-central limit theorems for long-range dependent series, and the limiting Hermite processes; fractional Brownian motion and its stochastic calculus; several celebrated decompositions of fractional Brownian motion; multidimensional models for long-range dependence and self-similarity; and maximum likelihood estimation methods for long-range dependent time series Designed for graduate students and researchers, each chapter of the book is supplemented by numerous exercises, some designed to test the reader’s understanding, while others invite the reader to consider some of the open research problems in the field today V L A D A S P I P I R A S is Professor of Statistics and Operations Research at the University of North Carolina, Chapel Hill M U R A D S TA Q Q U is Professor of Mathematics and Statistics at Boston University Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:11:08, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 C A M B R I D G E S E R I E S I N S TAT I S T I C A L A N D P RO BA B I L I S T I C M AT H E M AT I C S Editorial Board Z Ghahramani (Department of Engineering, University of Cambridge) R Gill (Mathematical Institute, Leiden University) F P Kelly (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge) B D Ripley (Department of Statistics, University of Oxford) S Ross (Department of Industrial and Systems Engineering, University of Southern California) M Stein (Department of Statistics, University of Chicago) This series of high-quality upper-division textbooks and expository monographs covers all aspects of stochastic applicable mathematics The topics range from pure and applied statistics to probability theory, operations research, optimization, and mathematical programming The books contain clear presentations of new developments in the field and also of the state of the art in classical methods While emphasizing rigorous treatment of theoretical methods, the books also contain applications and discussions of new techniques made possible by advances in computational practice A complete list of books in the series can be found at www.cambridge.org/statistics Recent titles include the following: 19 20 21 22 23 24 25 26 27 28 29 30 31 33 34 35 36 37 38 39 40 41 42 43 44 45 The Coordinate-Free Approach to Linear Models, by Michael J Wichura Random Graph Dynamics, by Rick Durrett Networks, by Peter Whittle Saddlepoint Approximations with Applications, by Ronald W Butler Applied Asymptotics, by A R Brazzale, A C Davison and N Reid Random Networks for Communication, by Massimo Franceschetti and Ronald Meester Design of Comparative Experiments, by R A Bailey Symmetry Studies, by Marlos A G Viana Model Selection and Model Averaging, by Gerda Claeskens and Nils Lid Hjort Bayesian Nonparametrics, edited by Nils Lid Hjort et al From Finite Sample to Asymptotic Methods in Statistics, by Pranab K Sen, Julio M Singer and Antonio C Pedrosa de Lima Brownian Motion, by Peter Mörters and Yuval Peres Probability (Fourth Edition), by Rick Durrett Stochastic Processes, by Richard F Bass Regression for Categorical Data, by Gerhard Tutz Exercises in Probability (Second Edition), by Loïc Chaumont and Marc Yor Statistical Principles for the Design of Experiments, by R Mead, S G Gilmour and A Mead Quantum Stochastics, by Mou-Hsiung Chang Nonparametric Estimation under Shape Constraints, by Piet Groeneboom and Geurt Jongbloed Large Sample Covariance Matrices and High-Dimensional Data Analysis, by Jianfeng Yao, Shurong Zheng and Zhidong Bai Mathematical Foundations of Infinite-Dimensional Statistical Models, by Evarist Giné and Richard Nickl Confidence, Likelihood, Probability, by Tore Schweder and Nils Lid Hjort Probability on Trees and Networks, by Russell Lyons and Yuval Peres Random Graphs and Complex Networks (Volume 1), by Remco van der Hofstad Fundamentals of Nonparametric Bayesian Inferences, by Subhashis Ghosal and Aad van der Vaart Long-Range Dependence and Self-Similarity, by Vladas Pipiras and Murad S Taqqu Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:11:08, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 Long-Range Dependence and Self-Similarity Vladas Pipiras University of North Carolina, Chapel Hill Murad S Taqqu Boston University Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:11:08, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence www.cambridge.org Information on this title: www.cambridge.org/9781107039469 DOI: 10.1017/9781139600347 c Vladas Pipiras and Murad S Taqqu 2017 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2017 Printed in the United States of America by Sheridan Books, Inc A catalog record for this publication is available from the British Library ISBN 978-1-107-03946-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:11:08, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 To Natércia and Filipa and to Rachelle, Yael, Jonathan, Noah, Kai and Olivia Downloaded from https:/www.cambridge.org/core New York University, on 02 Jun 2017 at 16:13:15, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 Downloaded from https:/www.cambridge.org/core New York University, on 02 Jun 2017 at 16:13:15, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 Contents List of Abbreviations Notation Preface page xv xvii xxi A Brief Overview of Time Series and Stochastic Processes Basics of Long-Range Dependence and Self-Similarity Physical Models for Long-Range Dependence and Self-Similarity 113 Hermite Processes 229 Non-Central and Central Limit Theorems 282 Fractional Calculus and Integration of Deterministic Functions with Respect to FBM 345 Stochastic Integration with Respect to Fractional Brownian Motion 397 Series Representations of Fractional Brownian Motion 437 Multidimensional Models 466 10 Maximum Likelihood Estimation Methods 539 A Auxiliary Notions and Results 575 B Integrals with Respect to Random Measures 588 C Basics of Malliavin Calculus 602 D Other Notes and Topics 610 Bibliography Index 15 613 660 vii Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:11:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347.001 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:11:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347.001 Expanded Contents List of Abbreviations Notation Preface xv xvii xxi 1.1 1 4 7 9 11 12 14 14 A Brief Overview of Time Series and Stochastic Processes Stochastic Processes and Time Series 1.1.1 1.1.2 1.1.3 1.2 Time Domain Perspective 1.2.1 1.3 Representations in the Time Domain Spectral Domain Perspective 1.3.1 1.3.2 1.3.3 1.3.4 1.4 Gaussian Stochastic Processes Stationarity (of Increments) Weak or Second-Order Stationarity (of Increments) Spectral Density Linear Filtering Periodogram Spectral Representation Integral Representations Heuristics 1.4.1 Representations of a Gaussian Continuous-Time Process 1.5 1.6 A Heuristic Overview of the Next Chapter Bibliographical Notes 2.1 2.2 Basics of Long-Range Dependence and Self-Similarity Definitions of Long-Range Dependent Series Relations Between the Various Definitions of Long-Range Dependence 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.3 2.4 Some Useful Properties of Slowly and Regularly Varying Functions Comparing Conditions II and III Comparing Conditions II and V Comparing Conditions I and II Comparing Conditions II and IV Comparing Conditions I and IV Comparing Conditions IV and III Comparing Conditions IV and V Short-Range Dependent Series and their Several Examples Examples of Long-Range Dependent Series: FARIMA Models 2.4.1 FARIMA(0, d, 0) Series 15 16 19 19 21 21 23 25 28 29 29 30 35 35 ix Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:14:16, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 x Expanded Contents 2.4.2 2.5 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.6.6 2.6.7 2.6.8 2.7 2.8 2.9 First Definition of LRD Under Heavy Tails: Condition A Second Definition of LRD Under Heavy Tails: Condition B Third Definition of LRD Under Heavy Tails: Codifference Heuristic Methods of Estimation 2.10.1 2.10.2 2.10.3 2.10.4 2.11 Fractional Brownian Motion Bifractional Brownian Motion The Rosenblatt Process SαS Lévy Motion Linear Fractional Stable Motion Log-Fractional Stable Motion The Telecom Process Linear Fractional Lévy Motion The Lamperti Transformation Connections Between Long-Range Dependent Series and Self-Similar Processes Long- and Short-Range Dependent Series with Infinite Variance 2.9.1 2.9.2 2.9.3 2.10 FARIMA( p, d, q) Series Definition and Basic Properties of Self-Similar Processes Examples of Self-Similar Processes The R/S Method Aggregated Variance Method Regression in the Spectral Domain Wavelet-Based Estimation Generation of Gaussian Long- and Short-Range Dependent Series 2.11.1 Using Cholesky Decomposition 2.11.2 Using Circulant Matrix Embedding 2.12 2.13 Exercises Bibliographical Notes 3.1 3.2 3.3 Physical Models for Long-Range Dependence and Self-Similarity Aggregation of Short-Range Dependent Series Mixture of Correlated Random Walks Infinite Source Poisson Model with Heavy Tails 3.3.1 3.3.2 3.3.3 3.3.4 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 Model Formulation Workload Process and its Basic Properties Input Rate Regimes Limiting Behavior of the Scaled Workload Process Power-Law Shot Noise Model Hierarchical Model Regime Switching Elastic Collision of Particles Motion of a Tagged Particle in a Simple Symmetric Exclusion Model Power-Law Pólya’s Urn Random Walk in Random Scenery Two-Dimensional Ising Model 3.11.1 Model Formulation and Result 42 43 47 47 53 56 59 59 61 62 63 65 67 76 76 82 82 84 84 88 88 93 99 100 100 106 108 113 113 117 120 120 123 128 131 149 154 156 162 167 172 177 180 181 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:14:16, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 ... Series and Stochastic Processes Basics of Long-Range Dependence and Self-Similarity Physical Models for Long-Range Dependence and Self-Similarity 113 Hermite Processes 229 Non-Central and Central... specialized and advanced topics on long-range dependence and self-similarity Chapter concerns physical models that give rise to long-range dependence and/ or self-similarity Chapters and focus on... Bibliographical Notes 2.1 2.2 Basics of Long-Range Dependence and Self-Similarity Definitions of Long-Range Dependent Series Relations Between the Various Definitions of Long-Range Dependence 2.2.1 2.2.2 2.2.3