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E-raamat: Long-Range Dependence and Self-Similarity

(University of North Carolina, Chapel Hill), (Boston University)
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Long-Range Dependence and Self-SimilaritySuurem pilt
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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.

Real-world time series rarely satisfy simple assumptions, often exhibiting long-range dependence. Ignoring this undermines accurate detection of trends and other important behavior. This text for graduate students and researchers in statistics and probability is also a reference for specialists in fields such as economics, finance, and hydrology.

Arvustused

'This is a marvelous book that brings together both classical background material and the latest research results on long-range dependence. The book is written so that it can be used as a main source by a graduate student, including all the essential proofs. I highly recommend this book.' Mark M. Meerschaert, Michigan State University 'This volume lays a rock-solid foundation for the subjects of long-range dependence and self-similarity. It also provides an up-to-date survey of more specialized topics at the center of this research area. The text is very readable and suitable for graduate courses, as it is self-contained and does not require more than an introductory course on stochastic calculus and time series. It is also written with the necessary level of mathematical detail to make it suitable for self-study. I particularly enjoyed the very nice introduction to fractional Brownian motion, its different representations, its stochastic calculus, and the connection to fractional calculus. I strongly recommend this book, which is a welcome addition to the literature and useful for a large audience.' Eric Moulines, Centre de Mathématiques Appliquées, École Polytechnique, Paris 'This book provides a modern, rigorous introduction to long-range dependence and self-similarity. The authors write with wonderful clarity, covering fundamental as well as selected specialized topics. The book can be highly recommended to anybody interested in mathematical foundations of long memory and self-similar processes.' Jan Beran, University of Konstanz, Germany 'This is the most readable and lucid account I have seen on long-range dependence and self-similarity. Pipiras and Taqqu present a time-series-centric view of this subject that should appeal to both practitioners and researchers in stochastic processes and statistics. I was especially enamored by the insightful comments on the history of the subject that conclude each chapter. This alone is worth the price of the book!' Richard Davis, Columbia University, New York

Muu info

A modern and rigorous introduction to long-range dependence and self-similarity, complemented by numerous more specialized up-to-date topics in this research area.
List of Abbreviations xv
Notation xvii
Preface xxi
1 A Brief Overview of Time Series and Stochastic Processes 1(14)
1.1 Stochastic Processes and Time Series
1(3)
1.1.1 Gaussian Stochastic Processes
2(1)
1.1.2 Stationarity (of Increments)
3(1)
1.1.3 Weak or Second-Order Stationarity (of Increments)
4(1)
1.2 Time Domain Perspective
4(3)
1.2.1 Representations in the Time Domain
4(3)
1.3 Spectral Domain Perspective
7(4)
1.3.1 Spectral Density
7(1)
1.3.2 Linear Filtering
8(1)
1.3.3 Periodogram
9(1)
1.3.4 Spectral Representation
9(2)
1.4 Integral Representations Heuristics
11(3)
1.4.1 Representations of a Gaussian Continuous-Time Process
12(2)
1.5 A Heuristic Overview of the Next
Chapter
14(1)
1.6 Bibliographical Notes
14(1)
2 Basics of Long-Range Dependence and Self-Similarity 15(98)
2.1 Definitions of Long-Range Dependent Series
16(3)
2.2 Relations Between the Various Definitions of Long-Range Dependence
19(11)
2.2.1 Some Useful Properties of Slowly and Regularly Varying Functions
19(2)
2.2.2 Comparing Conditions II and III
21(1)
2.2.3 Comparing Conditions II and V
21(2)
2.2.4 Comparing Conditions I and II
23(2)
2.2.5 Comparing Conditions II and IV
25(3)
2.2.6 Comparing Conditions I and IV
28(1)
2.2.7 Comparing Conditions IV and III
29(1)
2.2.8 Comparing Conditions IV and V
29(1)
2.3 Short-Range Dependent Series and their Several Examples
30(5)
2.4 Examples of Long-Range Dependent Series: FARIMA Models
35(8)
2.4.1 FARIMA(0, d, 0) Series
35(7)
2.4.2 FARIMA(p, d, q) Series
42(1)
2.5 Definition and Basic Properties of Self-Similar Processes
43(4)
2.6 Examples of Self-Similar Processes
47(18)
2.6.1 Fractional Brownian Motion
47(6)
2.6.2 Bifractional Brownian Motion
53(3)
2.6.3 The Rosenblatt Process
56(3)
2.6.4 SalphaS Levy Motion
59(1)
2.6.5 Linear Fractional Stable Motion
59(2)
2.6.6 Log-Fractional Stable Motion
61(1)
2.6.7 The Telecom Process
62(1)
2.6.8 Linear Fractional Levy Motion
63(2)
2.7 The Lamperti Transformation
65(2)
2.8 Connections Between Long-Range Dependent Series and Self-Similar Processes
67(9)
2.9 Long- and Short-Range Dependent Series with Infinite Variance
76(8)
2.9.1 First Definition of LRD Under Heavy Tails: Condition A
76(6)
2.9.2 Second Definition of LRD Under Heavy Tails: Condition B
82(1)
2.9.3 Third Definition of LRD Under Heavy Tails: Codifference
82(2)
2.10 Heuristic Methods of Estimation
84(15)
2.10.1 The R/S Method
84(4)
2.10.2 Aggregated Variance Method
88(1)
2.10.3 Regression in the Spectral Domain
88(5)
2.10.4 Wavelet-Based Estimation
93(6)
2.11 Generation of Gaussian Long- and Short-Range Dependent Series
99(7)
2.11.1 Using Cholesky Decomposition
100(1)
2.11.2 Using Circulant Matrix Embedding
100(6)
2.12 Exercises
106(2)
2.13 Bibliographical Notes
108(5)
3 Physical Models for Long-Range Dependence and Self-Similarity 113(116)
3.1 Aggregation of Short-Range Dependent Series
113(4)
3.2 Mixture of Correlated Random Walks
117(3)
3.3 Infinite Source Poisson Model with Heavy Tails
120(29)
3.3.1 Model Formulation
120(3)
3.3.2 Workload Process and its Basic Properties
123(5)
3.3.3 Input Rate Regimes
128(3)
3.3.4 Limiting Behavior of the Scaled Workload Process
131(18)
3.4 Power-Law Shot Noise Model
149(5)
3.5 Hierarchical Model
154(2)
3.6 Regime Switching
156(6)
3.7 Elastic Collision of Particles
162(5)
3.8 Motion of a Tagged Particle in a Simple Symmetric Exclusion Model
167(5)
3.9 Power-Law Polya's Urn
172(5)
3.10 Random Walk in Random Scenery
177(3)
3.11 Two-Dimensional Ising Model
180(35)
3.11.1 Model Formulation and Result
181(3)
3.11.2 Correlations, Dimers and Pfaffians
184(14)
3.11.3 Computation of the Inverse
198(11)
3.11.4 The Strong Szego Limit Theorem
209(3)
3.11.5 Long-Range Dependence at Critical Temperature
212(3)
3.12 Stochastic Heat Equation
215(1)
3.13 The Weierstrass Function Connection
216(5)
3.14 Exercises
221(2)
3.15 Bibliographical Notes
223(6)
4 Hermite Processes 229(53)
4.1 Hermite Polynomials and Multiple Stochastic Integrals
229(3)
4.2 Integral Representations of Hermite Processes
232(9)
4.2.1 Integral Representation in the Time Domain
232(1)
4.2.2 Integral Representation in the Spectral Domain
233(1)
4.2.3 Integral Representation on an Interval
234(5)
4.2.4 Summary
239(2)
4.3 Moments, Cumulants and Diagram Formulae for Multiple Integrals
241(13)
4.3.1 Diagram Formulae
241(4)
4.3.2 Multigraphs
245(1)
4.3.3 Relation Between Diagrams and Multigraphs
246(5)
4.3.4 Diagram and Multigraph Formulae for Hermite Polynomials
251(3)
4.4 Moments and Cumulants of Hermite Processes
254(5)
4.5 Multiple Integrals of Order Two
259(3)
4.6 The Rosenblatt Process
262(2)
4.7 The Rosenblatt Distribution
264(8)
4.8 CDF of the Rosenblatt Distribution
272(1)
4.9 Generalized Hermite and Related Processes
272(7)
4.10 Exercises
279(1)
4.11 Bibliographical Notes
280(2)
5 Non-Central and Central Limit Theorems 282(63)
5.1 Nonlinear Functions of Gaussian Random Variables
282(3)
5.2 Hermite Rank
285(3)
5.3 Non-Central Limit Theorem
288(10)
5.4 Central Limit Theorem
298(7)
5.5 The Fourth Moment Condition
305(1)
5.6 Limit Theorems in the Linear Case
306(10)
5.6.1 Direct Approach for Entire Functions
306(5)
5.6.2 Approach Based on Martingale Differences
311(5)
5.7 Multivariate Limit Theorems
316(12)
5.7.1 The SRD Case
316(3)
5.7.2 The LRD Case
319(2)
5.7.3 The Mixed Case
321(3)
5.7.4 Multivariate Limits of Multilinear Processes
324(4)
5.8 Generation of Non-Gaussian Long- and Short-Range Dependent Series
328(13)
5.8.1 Matching a Marginal Distribution
329(2)
5.8.2 Relationship Between Autocorrelations
331(2)
5.8.3 Price Theorem
333(3)
5.8.4 Matching a Targeted Autocovariance for Series with Prescribed Marginal
336(5)
5.9 Exercises
341(1)
5.10 Bibliographical Notes
342(3)
6 Fractional Calculus and Integration of Deterministic Functions with Respect to FBM 345(52)
6.1 Fractional Integrals and Derivatives
345(14)
6.1.1 Fractional Integrals on an Interval
345(3)
6.1.2 Riemann-Liouville Fractional Derivatives D on an Interval
348(4)
6.1.3 Fractional Integrals and Derivatives on the Real Line
352(2)
6.1.4 Marchaud Fractional Derivatives D on the Real Line
354(3)
6.1.5 The Fourier Transform Perspective
357(2)
6.2 Representations of Fractional Brownian Motion
359(10)
6.2.1 Representation of FBM on an Interval
359(9)
6.2.2 Representations of FBM on the Real Line
368(1)
6.3 Fractional Wiener Integrals and their Deterministic Integrands
369(17)
6.3.1 The Gaussian Space Generated by Fractional Wiener Integrals
369(3)
6.3.2 Classes of Integrands on an Interval
372(5)
6.3.3 Subspaces of Classes of Integrands
377(4)
6.3.4 The Fundamental Martingale
381(1)
6.3.5 The Deconvolution Formula
382(1)
6.3.6 Classes of Integrands on the Real Line
383(1)
6.3.7 Connection to the Reproducing Kernel Hilbert Space
384(2)
6.4 Applications
386(8)
6.4.1 Girsanov's Formula for FBM
386(2)
6.4.2 The Prediction Formula for FBM
388(4)
6.4.3 Elementary Linear Filtering Involving FBM
392(2)
6.5 Exercises
394(1)
6.6 Bibliographical Notes
395(2)
7 Stochastic Integration with Respect to Fractional Brownian Motion 397(40)
7.1 Stochastic Integration with Random Integrands
397(16)
7.1.1 FBM and the Semimartingale Property
397(2)
7.1.2 Divergence Integral for FBM
399(3)
7.1.3 Self-Integration of FBM
402(5)
7.1.4 Ito's Formulas
407(6)
7.2 Applications of Stochastic Integration
413(21)
7.2.1 Stochastic Differential Equations Driven by FBM
413(1)
7.2.2 Regularity of Laws Related to FBM
414(4)
7.2.3 Numerical Solutions of SDEs Driven by FBM
418(8)
7.2.4 Convergence to Normal Law Using Stein's Method
426(4)
7.2.5 Local Time of FBM
430(4)
7.3 Exercises
434(1)
7.4 Bibliographical Notes
435(2)
8 Series Representations of Fractional Brownian Motion 437(29)
8.1 Karhunen-Loeve Decomposition and FBM
437(3)
8.1.1 The Case of General Stochastic Processes
437(1)
8.1.2 The Case of BM
438(1)
8.1.3 The Case of FBM
439(1)
8.2 Wavelet Expansion of FBM
440(15)
8.2.1 Orthogonal Wavelet Bases
440(5)
8.2.2 Fractional Wavelets
445(5)
8.2.3 Fractional Conjugate Mirror Filters
450(2)
8.2.4 Wavelet-Based Expansion and Simulation of FBM
452(3)
8.3 Paley-Wiener Representation of FBM
455(8)
8.3.1 Complex-Valued FBM and its Representations
455(1)
8.3.2 Space La and its Orthonormal Basis
456(6)
8.3.3 Expansion of FBM
462(1)
8.4 Exercises
463(1)
8.5 Bibliographical Notes
464(2)
9 Multidimensional Models 466(73)
9.1 Fundamentals of Multidimensional Models
467(5)
9.1.1 Basics of Matrix Analysis
467(2)
9.1.2 Vector Setting
469(2)
9.1.3 Spatial Setting
471(1)
9.2 Operator Self-Similarity
472(3)
9.3 Vector Operator Fractional Brownian Motions
475(20)
9.3.1 Integral Representations
476(8)
9.3.2 Time Reversible Vector OFBMs
484(2)
9.3.3 Vector Fractional Brownian Motions
486(6)
9.3.4 Identifiability Questions
492(3)
9.4 Vector Long-Range Dependence
495(13)
9.4.1 Definitions and Basic Properties
495(4)
9.4.2 Vector FARIM A (0, D, 0) Series
499(4)
9.4.3 Vector PGN Series
503(1)
9.4.4 Fractional Cointegration
504(4)
9.5 Operator Fractional Brownian Fields
508(18)
9.5.1 M-Homogeneous Functions
509(4)
9.5.2 Integral Representations
513(10)
9.5.3 Special Subclasses and Examples of OFBFs
523(3)
9.6 Spatial Long-Range Dependence
526(6)
9.6.1 Definitions and Basic Properties
526(3)
9.6.2 Examples
529(3)
9.7 Exercises
532(3)
9.8 Bibliographical Notes
535(4)
10 Maximum Likelihood Estimation Methods 539(36)
10.1 Exact Gaussian MLE in the Time Domain
539(3)
10.2 Approximate MLE
542(9)
10.2.1 Whittle Estimation in the Spectral Domain
542(8)
10.2.2 Autoregressive Approximation
550(1)
10.3 Model Selection and Diagnostics
551(3)
10.4 Forecasting
554(1)
10.5 R Packages and Case Studies
555(3)
10.5.1 The ARFIMA Package
555(1)
10.5.2 The FRACDIFF Package
556(2)
10.6 Local Whittle Estimation
558(9)
10.6.1 Local Whittle Estimator
558(4)
10.6.2 Bandwidth Selection
562(3)
10.6.3 Bias Reduction and Rate Optimality
565(2)
10.7 Broadband Whittle Approach
567(2)
10.8 Exercises
569(1)
10.9 Bibliographical Notes
570(5)
A Auxiliary Notions and Results 575(13)
A1 Fourier Series and Fourier Transforms
575(4)
A1.1 Fourier Series and Fourier Transform for Sequences
575(2)
A1.2 Fourier Transform for Functions
577(2)
A2 Fourier Series of Regularly Varying Sequences
579(4)
A3 Weak and Vague Convergence of Measures
583(2)
A3.1 The Case of Probability Measures
583(1)
A3.2 The Case of Locally Finite Measures
584(1)
A4 Stable and Heavy-Tailed Random Variables and Series
585(3)
B Integrals with Respect to Random Measures 588(14)
B1 Single Integrals with Respect to Random Measures
588(9)
B1.1 Integrals with Respect to Random Measures with Orthogonal Increments
589(1)
B1.2 Integrals with Respect to Gaussian Measures
590(2)
B1.3 Integrals with Respect to Stable Measures
592(1)
B1.4 Integrals with Respect to Poisson Measures
593(3)
B1.5 Integrals with Respect to Levy Measures
596(1)
B2 Multiple Integrals with Respect to Gaussian Measures
597(5)
C Basics of Malliavin Calculus 602(8)
C1 Isonormal Gaussian Processes
602(1)
C2 Derivative Operator
603(4)
C3 Divergence Integral
607(1)
C4 Generator of the Ornstein-Uhlenbeck Semigroup
608(2)
D Other Notes and Topics 610(3)
Bibliography 613(47)
Index 660
Vladas Pipiras is Professor of Statistics and Operations Research at the University of North Carolina, Chapel Hill. His research focuses on stochastic processes exhibiting long-range dependence, self-similarity, and other scaling phenomena, as well as on stable, extreme-value and other distributions possessing heavy tails. His other current interests include high-dimensional time series, sampling issues for 'big data', and stochastic dynamical systems, with applications in econometrics, neuroscience, engineering, computer science, and other areas. He has written over fifty research papers and is coauthor of A Basic Course in Measure and Probability: Theory for Applications (with Ross Leadbetter and Stamatis Cambanis, Cambridge, 2014) Murad S. Taqqu's research involves self-similar processes, their connection to time series with long-range dependence, the development of statistical tests, and the study of non-Gaussian processes whose marginal distributions have heavy tails. He has written more than 250 scientific papers and is coauthor of Stable Non-Gaussian Random Processes (with Gennady Samorodnitsky, 1994). Professor Taqqu is a Fellow of the Institute of Mathematical Statistics and has been elected Member of the International Statistical Institute. He has received a number of awards, including a John Simon Guggenheim Fellowship, the 1995 William J. Bennett Award, the 1996 Institute of Electrical and Electronics Engineers W. R. G. Baker Prize, the 2002 EURASIP Best Paper in Signal Processing Award, and the 2006 Association for Computing Machinery Special Interest Group on Data Communications (ACM SIGCOMM) Test of Time Award.
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