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 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Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:27, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347.017 Index 1/ f noise, 227 SαS Lévy motion, 59, 60 convergence to, 166 Abelian theorem, 109 aggregated variance method, 88 aggregation of SRD series, 113 beta-like mixture distribution, 116 disaggregation problem, 117 long-range dependence, 115 mixture distribution, 117 anomalous diffusion, 227, 396 antipersistent series, 31, 38, 40, 72, 109 Appell’s hypergeometric function, 390 applications of LRD, 572, 610 autocorrelation function (ACF), autocovariance function (ACVF), AR series, FARIMA(0, d, 0) series, 37 FARIMA( p, d, q) series, 43, 569, 570 FGN, 23, 67 function of Gaussian series, 285 MA series, spatial setting, 471 vector FARIMA(0, D, 0) series, 500 vector setting, 469 white noise, autoregressive (AR) series, 5, 8, 9, 339 aggregation, 114 autoregressive moving average (ARMA) series, 6, 31, 570 characteristic polynomial, 32 short-range dependence, 31 backshift operator, 31, 357, 531 forward-shift operator, 499 Barnes G–function, 212 Bessel function of the first kind, 458 beta function, 24 bifractional Brownian motion (biFBM), 53, 110, 216 Boltzmann (Gibbs) distribution, see Ising model bridge process, 86 Brownian bridge, 87 Brownian motion convergence to, 146 Brownian motion (BM), 2, 49, 54, 59, 62, 166 convergence to, 118, 158, 221, 298, 316 integral representations, 13 integrated, 166 Karhunen–Loève decomposition, 438 Lamperti transformation, 66 Paley–Wiener expansion, 456, 462 two-parameter, 215 wavelet-based expansion (midpoint displacement), 452 Wiener integral, 371 Campbell’s theorem, 125 Caputo, see fractional derivative Cauchy determinant, 213 Cauchy’s integral formula, 206 central limit theorem, 298 fourth moment condition, 305 functional of Markov chains, 161 Lindeberg–Feller, 92 martingale, 177 multilinear processes, 327 multivariate extension, 316 multivariate extension (mixed case), 322 characteristic polynomial ARMA series, 32 matrix, 468 Cholesky, see generation of Gaussian series circulant matrix, 100, 201 eigenvectors and eigenvalues, 100 codifference, 83 completely monotone function, 530, 535 contraction, 305, 323, 600 convergence conditional, 25 Cramér–Wold device, 584 in L ( )–sense, in distribution, of random variables, vectors and processes, 584 660 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 Index vague (locally weak), of locally finite measures, 584 weak, of locally finite measures, 584 weak, of probability measures, 583 convergence to types theorem, 68 convolution Fourier transform, 576, 577 functions, 577 sequences, 7, 576 correlated random walk (CRW), 117, 162, 173 covariance function, biFBM, 53 covariance function, 216 FBM, 47 OU process, 13 VFBM, 490 cumulant, 244 joint, 244 diagram association with multigraphs, 247 connected, 244 definition, 241 formula for cumulants of Hermite polynomials of Gaussian variables, 253 formula for cumulants of multiple integrals, 245 formula for moments of Hermite polynomials of Gaussian variables, 251 formula for moments of multiple integrals (spectral domain), 244 formula for moments of multiple integrals (time domain), 242 Gaussian, 242 non-flat, 242 difference (differencing) operator, 3, 36 digamma function, 98 discrete-chaos process, see multilinear process Durbin–Levinson algorithm, 107, 541 elastic collision of particles, 162, 168 entire function, 306 ergodic theorem, 398, 405 estimation in LRD series aggregated variance method, 88 Bayesian, 573 bootstrap, 574 impulse response coefficients, 573 KPSS method, 111 maximum likelihood, see maximum likelihood estimation (MLE) R/S method, 84 regression in spectral domain (GPH estimation), 88 regression with LRD errors, 573 resampling (bootstrap), 574 seasonal models, 573 self-normalization, 574 state space modeling, 573 661 subsampling, 574 V/S method, 111 wavelet-based, 93 estimation of ACVF and ACF, 574 estimation of mean, 540 best linear unbiased estimator (BLUE), 540, 569 Euler’s constant, 214 exponent rank, 306 FARIMA models, 35 data applications, 555 disaggregation, 117 equivalence of LRD conditions, 43 FARIMA(0, d, 0) series, 35, 542, 550 FARIMA( p, d, q) series, 42, 539 generation, 105 heavy-tailed, 82, 573 invertibility, 38 principal range, 41 spatial FARIMA, 530–532 time plots, 40, 82 vector FARIMA series, 499 wavelet-based expansion of FBM, 449 fast Fourier transform (FFT), 9, 61, 101, 524, 544 fast wavelet transform (FWT), 97, 442, 454 complexity, 443 fast biorthogoal wavelet transform, 450 Fernique’s inequality, 409, 422 filter high-pass, 441 linear, 8, 33, 39 low-pass, 441 finite-dimensional distributions, Fourier frequencies, 9, 558 Fourier series ACVF and spectral density, 576 convolution, 577 definition, 575 Fourier coefficients, 575 Fourier transform, 576 Parseval’s identity, 576 regularly varying sequences, 579, 583 Fourier transform (functions) convolution, 578 definition, 577 extensions, 578 inversion formula, 577 Parseval’s identity, 577 Plancherel’s identity, 577 fourth moment condition, 305 fractional Brownian motion (FBM) pth variation, 398 arbitrage, 435 complex-valued, 455 convergence to, 64, 71, 72, 119, 132, 152, 164, 166, 172, 177, 220 deconvolution formula, 383 definition, 47 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 662 Fernique’s inequality, 409, 422 Gaussian space, 369 geometric, 412, 425 Girsanov’s formula, 387 Hölder continuity, 398 intersection local time, 436 Karhunen–Loève decomposition, 439 kernel function on interval, 359, 361, 363 large deviations, 611 linear filtering, 392 linear span and its closure, 369 local time, 430, 433 not semimartingale, 398 operator field, see operator fractional Brownian field (OFBF) Paley–Wiener expansion, 462 parametrization, 359 prediction formula, 389 quadratic variation, 427 realizations, 53 representation on interval, 364 RKHS, 385 spectral domain representation, 51, 368 standard, 47 tempered, 395 time domain representation, 48, 49, 63, 368 types I and II, 110 vector operator, see vector operator fractional Brownian motion (vector OFBM) wavelet-based expansion, 452 wavelet-based simulation, 454 fractional Brownian sheet (FBS), 465, 525 fractional cointegration cointegrating rank, 505 cointegrating spaces, 507 cointegrating vector, 504 definition, 504 fractional cointegration model, 506 fractionally cointegrated, 504 vector OFBM, 507 fractional derivative Caputo, 352 composition with fractional integral, 350, 356 exponential function, 353 Fourier transform, 358 function of two variables, 400 Grünwald–Letnikov, 359 in representation of FBM, 364, 368 indicator functions, 353, 356, 358 Liouville (on real line), 352, 356 Marchaud (on real line), 355, 356 partie finie (finite part), 356 power functions, 350 renormalization, 355, 357 Riemann–Liouville (on interval), 349 weighted, 375 fractional diffusion equation, 396 fractional Gaussian noise (FGN), 23, 68, 108 Index ergodic, 398 vector, 503, 508 fractional integral composition with fractional derivative, 350 exponential function, 353 Fourier transform, 358 fractional integration by parts, 347, 354 function of two variables, 400 in representation of FBM, 364, 368 indicator functions, 353, 358 Liouville (on real line), 352 power functions, 347 reflection property, 347, 354 Riemann–Liouville (on interval), 346 semigroup property, 347, 354 fractional Ornstein–Uhlenbeck (OU) process, 384, 394 fractional Wiener integral class of integrands, 370 classes of integrands for κ ∈ (−1/2, 0), 375, 376, 379 classes of integrands for κ ∈ (0, 1/2), 372, 377, 378 classes of integrands on real line, 383 completeness, 370–372, 375, 377, 378 connection to RKHS, 384 covariance, 374, 377 deconvolution formula, 383 definition, 370 definition for κ ∈ (−1/2, 0), 375 definition for κ ∈ (0, 1/2), 374 elementary (step) function, 370 fractional OU process, 384, 394 fundamental martingale, 381, 386, 456 Gaussian space, 369 Girsanov’s formula, 387 linear filtering, 392 prediction formula, 389 Fubini-based argument, 478 functions of Gaussian variables, 282 covariance, 284, 331 exponent rank, 306 Hermite rank, 285 long-range dependence, 286 matching marginal distribution, 329 power rank, 313 gamma function, 24 Gauss hypergeometric function, 266, 363 Gaussian subordination, see functions of Gaussian variables Gegenbauer ARMA (GARMA) models, 109 G frequency (Gegenbauer frequency), 110 generalized dominated convergence theorem, 80 generalized Hermite process, 276 kernel, 274 generalized inverse, 129, 329 regularly varying function, 132, 586 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 Index generalized Riemann zeta function, 98 generation of Gaussian series Cholesky method, 100 circulant matrix embedding method, 100 extensions, 112 generation of non-Gaussian series, 328 circulant matrix embedding, 337 matching autocovariance, 336 matching marginal distribution, 329 Price theorem, 333 geometric FBM, 412, 425 Girsanov’s formula, 387 GPH estimation estimator, 89, 560, 573 Hölder continuity, 379 Hölder norm, 424 heavy-tailed random variable definition, 586 infinite source Poisson model, 121 heavy-tailed time series definition, 586 FARIMA series, 82 linear, 77 long-range dependence, 76, 82 short-range dependence, 77, 82 Hermite expansion, 284 coefficients, 284, 341 exponential function, 331 Hermite rank, 285 indicator function, 332 Hermite polynomial connection to multiple integrals, 230 definition, 229 recursion property, 231, 279 Hermite process convergence to, 288, 307, 313 cumulants, 258 generalized, 276 moments, 254 representation on interval, 234, 240 representation on positive half-axis, 237, 239, 240 spectral domain representation, 234, 239, 240 time domain representation, 232, 239, 240 Hermite rank, 285 Hermitian, see matrix analysis Hermitian function, 599 hierarchical model, 154 Hilbert space, 370 Hilbert–Schmidt theorem, 438 homogeneous function, 509 Hurst index (parameter), 44 inequality generalized Minkowski’s, 353 geometric and arithmetic means, 303 Young’s, 430 663 infinite source Poisson model, 120 convergence to BM, 146 convergence to FBM, 132 convergence to intermediate Telecom process, 144 convergence to stable Lévy motion, 140, 147 convergence to Telecom process, 133 input rate regimes (fast, slow and intermediate), 129 long-range dependence, 128 related models, 224 workload process, 123 infrared correction, 49, 217 inner product space, 370 complete, 370 innovations algorithm, 107, 541 integral representation, 11 integrated series, Ising model, 180 critical temperature, 182 Boltzmann (Gibbs) distribution, 182 boundary conditions, 183 configurations, 182 counting lattice, 187 Curie temperature, 183 dimer, 189 energy (Hamiltonian), 182 external magnetic field, 182 ferromagnetism, 183 inverse temperature, 182 Ising lattice, 181 long-range dependence, 183 one-dimensional, 223 Pfaffian, 189, 192 phase transition, 182 spins, 183 strong Szegö limit theorem, 209 temperature, 182 thermodynamic limit, 181 Itô’s formulas, 407, 433 Itô–Nisio theorem, 463 Karamata’s theorem, 20 Karhunen–Loève decomposition, 438 BM, 438 FBM, 439 Kolmogorov’s continuity criterion, 398 Kolmogorov’s formula, 544 Lévy, see random measure Lamperti transformation, 66 BM, 66 Lamperti’s limit theorem, 69 vector setting, 487 Langevin stochastic differential equation, 3, 384 large deviations of FBM, 611 Leibniz’s rule for differentiation under integral sign, 349 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 664 Index linear filter, 8, 33, 39 linear fractional Lévy motion, 63 linear fractional stable motion (LFSM) convergence to, 65, 78 definition, 60 integral representation, 59 linear fractional stable noise (LFSN), 77, 83 linear time series, heavy-tailed, 77 Liouville, see fractional derivative, see fractional integral local time, 178, 430 local Whittle estimation bandwidth selection, 562 bias reduction, 565 bias-variance tradeoff, 563 estimator, 559 GPH estimator, 560 large sample asymptotics, 560 local polynomial estimator, 566 mean squared error (MSE), 564 non-smooth models, 564 optimal bandwidth, 565 optimal rate, 566 rate optimality, 565 smooth models, 564 tapering, 565 Whittle plot, 562 log-fractional stable motion (log-FSM), 62, 108 long memory, see long-range dependence long-range dependence (LRD) conditions I-V, 17 connections to self-similar processes, 68 definition, 18 infinite variance, 76 long-run variance, 28 mixing, 612 non-linear time series, 610 nonstationarity, 573 origins, 108 parameter, 18 point processes, 610 spatial, see spatial long-range dependence (LRD) vector, see vector long-range dependence (LRD) long-run variance LRD series, 23, 28 SRD series, 31 Malliavin calculus chain rule, 415, 604 chaos expansion, 431, 605 derivative operator, 400, 603, 606 divergence (Skorokhod) integral, 399, 607 generator of OU semigroup, 426, 608 integration by parts formula, 604 isonormal Gaussian process, 399, 602 smooth random variables, 603 space D1, p , 604 1, p space Dloc , 604 space Dk, p , 605 Stroock formula, 431, 606 white noise case, 602, 603, 605, 606 Marchaud, see fractional derivative Markov chain, 33 β–regular, 228 central limit theorem, 161 short-range dependence, 33 Markov switching model, 156 convergence to BM, 158 matrix analysis centralizer, 494 characteristic polynomial, 468 diagonal matrix, 468 eigenvalue, 468 eigenvector, 468 Hermitian symmetric matrix, 468 Hermitian transpose, 467 Jordan block, 468 Jordan decomposition, 468, 477, 533 matrix exponential, 472, 533 matrix norm, 469 positive (semi)definite matrix, 468 primary matrix function, 480 singular, 468 square root of positive semidefinite matrix, 468, 478 maximum likelihood estimation (MLE) applications, 572 autoregressive approximation, 550 best linear predictor, 541 broadband Whittle, 567 Durbin–Levinson algorithm, 541 exact Gaussian MLE, 539 FARIMA(0, d, 0) series, 550 finite (small) samples, 550, 571 forecasting, 554 goodness of fit, 552 ill conditioning, 542 implementation, 570 information criteria (AIC, BIC, HIC), 552 innovations algorithm, 541 large sample asymptotics, 545, 551, 570 likelihood function, 540 local Whittle, see local Whittle estimation maximum likelihood estimator, 540 model selection, 551 portmanteau tests, 552 quadratic form, 547 R packages, 555, 571 spectral window, 553 temperature data, 555, 557 Whittle’s approximation and estimation, 542, 569 Mehler’s formula, 342 method of moments, 300, 317 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 Index mixing, 612 mixture of correlated random walks, 117 convergence to FBM, 119 moving average (MA) series, multifractal processes, 611 multifractional motions, 611 multigraph k–regular, 246 (non-)standard, 302 association with diagrams, 247 connected, 248 cumulants of Hermite process, 258 cumulants of multiple integral of order two, 259 definition, 246 degree, 246 formula for cumulants of Hermite polynomials of Gaussian variables, 253 formula for cumulants of multiple integrals, 250 formula for moments of Hermite polynomials of Gaussian variables, 253 formula for moments of multiple integrals, 250 moments of Hermite process, 254 multiplicity number, 246 pair sequence (set of lines), 246 multilinear process, 324 limit theorem, 327 multinomial occupancy scheme, 228 multiple integral of order two, 259 characteristic function, 262 cumulants, 259 series representation, 261 multiple integrals with respect to Gaussian measures change of variables formula, 599 connection between time and spectral domains, 600 connection to Itô stochastic integrals, 601 contraction, 600 covariance, 598 definition, 598 Hermitian case, 599 Hermitian function, 599 product, 600, 601 simple function, 597 stochastic Fubini theorem, 601 symmetrization, 598 tensor product, 600 multiresolution analysis (MRA), see wavelet analysis Nile river time series, 87, 93, 99, 108, 570 non-central limit theorem, 288 linear time series (entire functions), 307 linear time series (martingale differences), 313 multilinear processes, 327 multivariate extension, 320 multivariate extension (mixed case), 322 665 ON/OFF model, 224 operator fractional Brownian field (OFBF) admissible functions, 521 anisotropic fractional Brownian field (anisotropic FBF), 525 definition, 508 domain symmetry group, 509 fractional Brownian sheet (FBS), 525 homogeneous function, 509, 511 identifiability, 509 isotropic fractional Brownian field (isotropic FBF), 524 operator scaling, 508 polar coordinate representation, 510 sample path properties, 537 spectral domain representation, 513 symmetry group, 510 tangent space of symmetry group, 509 time domain representation, 516, 521 operator scaling Gaussian random field, see operator fractional Brownian field (OFBF) operator self-similarity, 472 operator stable distribution, 535 operator stable distribution, 535 order statistics, see simple symmetric exclusion Ornstein–Uhlenbeck (OU) process, 2, integral representations, 13 Lamperti transformation, 66 orthant probability for normal vector, 332 Pólya’s urn, 172 ancestral line, 173 convergence to FBM, 177 correlated random walk, 173 Paley–Wiener expansion of FBM, 462 Bessel function of the first kind, 458 BM, 456, 462 fundamental martingale, 456 isometric Hilbert spaces, 457 uniform convergence, 463 zeros of Bessel functions, 461 Parseval’s identity, 576, 578 partial autocorrelation function, 107 periodogram, Pfaffian, see Ising model Plancherel’s identity, 578 Pochhammer symbol, 363 Poisson, see random measure polar coordinate representation, see operator fractional Brownian field (OFBF) Potter’s bound, 19 Potters bound, 21 power rank, 313 Price theorem, 333 quadratic variation, 382, 398 quasi-monotonicity, 25, 497, 579 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 666 Index R/S method, 84 R/S plot (poxplot), 87 R/S statistic, 85 random field, 466 conditions for ACVF, 530 stationary, 471 random measure σ –additive, 588 compensated Poisson, 594 control measure, 588 Gaussian, 590 Hermitian Gaussian, 590 independently scattered, 588 Lévy, 596 Poisson, 593 stable, 592 with orthogonal increments, 589 random walk, in random environment, 178 in random scenery, 177 reduction theorem, 292 regime switching, 156 regression in spectral domain, see GPH estimation regularly varying function, 16 uniform convergence theorem, 19 renewal process, 224 renewal-reward process, 224 reproducing kernel Hilbert space (RKHS), 384 FBM, 385 reversion operation, 340 Riemann zeta function, 267 generalized, 267 Riemann–Liouville, see fractional integral, see fractional derivative Riemann–Stieltjes integral, 408 Rosenblatt distribution CDF, 270, 272 CDF plots, 272 CDF table, 273 characteristic function, 265 cumulants, 265 definition, 264 Lévy–Khintchine representation, 268 PDF, 270 PDF plots, 272 series representation, 266 Rosenblatt process characteristic function, 263 convergence to, 74 cumulants, 263 definition, 58, 263 integral representations, 57 Schramm–Loewner evolution (SLE), 227 self-similar Markov processes, 612 self-similar process asymptotically at infinity, 64, 145 basic properties, 45 definition, 43 locally asymptotically, 65, 145 operator, see operator self-similarity second-order, 144 self-similarity parameter, 44 semimartingale, 398 short-range dependence (SRD) ARMA series, 31 definition, 30 infinite variance, 76 Markov chain, 33 vector, 497 shot noise model, 149 convergence to FBM, 152 long-range dependence, 151 shots, 150 simple symmetric exclusion, 167 convergence to FBM, 172 extensions, 225 order statistics, 168 stirring motion, 168 tagged particle, 167 single integral with respect to random measure compensated Poisson measure, 595 connection for Lévy and Poisson measures, 597 connection for stable and Poisson measures, 596 Gaussian measure, 590 Lévy measure, 597 Poisson measure, 594 random measure with orthogonal increments, 590 simple (elementary) function, 589 spectral domain, 592 stable measure, 593 time domain, 592 slowly varying function, 16 Karamata’s theorem, 20 Potter’s bound, 19 quasi-monotone, 25 Zygmund class, 25 spatial long-range dependence (LRD) anisotropic, 528 Cauchy family, 530 condition IV-s-anisot, 528 conditions II-s-isot and IV-s-isot, 526 isotropic, 526 scaling transition, 529 spherical harmonics, 526 spatial process, see random field spectral density, AR series, 8, FARIMA(0, d, 0) series, 37 FARIMA( p, d, q) series, 43 FGN, 67 spatial setting, 471 vector setting, 469 white noise, spectral domain, Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 Index spectral measure, 10 spectral representation, stable Lévy motion, 123 transition probability, 396 stable random measure, 592 stable random variable definition, 585 domain of attraction, 585 symmetric α-stable (SαS), 585 stable self-similar processes with stationary increments, 611 stationarity fields, 471 strict, vector, 469 weak or second order, stationarity of increments, 3, 44 Stein’s method, 426 quadratic variation of FBM, 427 total variation distance, 426 Stirling’s formula, 36 stirring motion, see simple symmetric exclusion stochastic differential equation (SDE) for FBM, 413 geometric FBM, 425 derivative of solution, 414 Doss–Sussmann transformation, 413 ergodicity, 436 Euler scheme, 419 existence and uniqueness of solution, 413 modified Euler scheme, 419 numerical solutions, 418 rate of convergence of numerical solutions, 419 regularity of law of solution, 418 statistical inference, 436 stochastic Fubini theorem, see multiple integrals with respect to Gaussian measures stochastic heat equation, 215 biFBM, 216 stochastic integration for FBM, 397 chaos expansion of local time, 431 derivative of solution of SDE, 414 divergence integral, 399 domain of divergence integral, 400 Doss–Sussmann transformation, 413 extended domain of divergence integral, 406 fractional white noise, 435 isonormal Gaussian process, 399 Itô’s formula, 408, 409, 412, 434 Itô’s formula for convex functions, 433 Itô’s formula for geometric FBM, 412 moments of local time, 434 nonlinear filtering, 436 numerical solutions of SDE, 418 optimal control, 436 quadratic variation, 427 regularity of law of solution of SDE, 418 regularity of laws, 414 667 Riemann–Stieltjes integral, 408 rough paths, 435 self-integration, 402, 406 Stein’s method, 426 stochastic differential equation (SDE), 413 Stratonovich integral, 408 supremum of FBM, 434 Tanaka’s formula, 433 tensor product Hilbert space, 400, 401 Wick product, 435 stochastic partial differential equation, 215, 227 parabolic Anderson model, 227 stochastic process, Gaussian, linear, stationary (strictly), stationary (weakly or second order), stationary increment (strictly), 3, 44 strong Szegö limit theorem classical, 210 with singularity, 211 summation by parts formula, 579 supremum norm, 424 Szegö’s limit theorem, 543 tagged particle, see simple symmetric exclusion Tanaka’s formula, 433 tangent space, see vector operator fractional Brownian motion (vector OFBM) Tauberian theorem, 109 Telecom process convergence to, 133 definition, 62, 122 intermediate, 123 tensor product, 600 time domain, time lag, time reversibility, see vector operator fractional Brownian motion (vector OFBM) time series, linear, nonlinear, 610 Toeplitz matrix, 210 total variation distance, 426 uniform convergence theorem, 19 vague convergence, 292, 584 vector long-range dependence (LRD) amplitude parameters, 496 conditions II-v and IV-v, 495 definition, 496 fractal connectivity, 537 fractional cointegration, 504 linear representations, 502, 534 phase parameters, 496 trigonometric power-law coefficients, 502, 536 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms https://doi.org/10.1017/CBO9781139600347 668 Index vector operator fractional Brownian motion (vector OFBM) continuity in distribution, 476 convergence to, 536 definition, 475 fractional cointegration, 507 identifiability, 492 Lamperti’s theorem, 487 primary matrix function, 480 proper, 476 real spectrum, 495 sample path properties, 536 spectral domain representation, 476 symmetry group, 493 tangent space of symmetry group, 493 time domain representation, 481 time reversibility, 485 vector fractional Brownian motion (VFBM), 486 vector process, 466 vector-valued stationary series, 469 amplitude spectrum, 470 coherence, 470, 497 definitions, 470 phase spectrum, 470 time reversibility, 469 wavelet analysis approximation coefficients, 441 biorthogonal wavelet basis, 449 conjugate mirror filters (CMFs), 441 Daubechies MRA, 444 downsampling by operation, 442 expansion of BM, 452 expansion of FBM, 452 fast wavelet transform (FWT), 97, 442 fractional CMFs, 450 fractional scaling function, 448 fractional wavelet, 447 Haar MRA, 443 Meyer MRA, 445 multiresolution analysis (MRA), 443 orthonormal wavelet basis, 440 scaling function, 441 simulation of FBM, 454 upsampling by operation, 443 wavelet, 94, 441 wavelet (detail) coefficients, 94, 441 wavelet crime, 443 zero moments, 94, 442, 450 wavelet Whittle estimation, 574 wavelet-based estimation, 93 decorrelation property, 96 log-scale diagram, 99 Weierstrass function, 216 convergence to FBM, 220 Weirstrass–Mandelbrot process, 217 white noise (WN), 4, Wiener integral, 371 Wiener process, see Brownian motion Wiener–Hopf factorization, 211 Young’s inequality, 430 Zygmund class, 25, 527 Downloaded from https:/www.cambridge.org/core Columbia University Libraries, on 02 Jun 2017 at 16:27:00, 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