Long-range dependence and self-similarity / Vladas Pipiras, University of North Carolina, Chapel Hill, Murad S. Taqqu, Boston University
- Author:
- Pipiras, Vladas
- Published:
- Cambridge ; New York NY : Cambridge University Press, 2017.
- Physical Description:
- xxiii, 668 pages ; 26 cm.
- Additional Creators:
- Taqqu, Murad S.
- Series:
- Contents:
- Machine generated contents note: 1.1.Stochastic Processes and Time Series -- 1.1.1.Gaussian Stochastic Processes -- 1.1.2.Stationarity (of Increments) -- 1.1.3.Weak or Second-Order Stationarity (of Increments) -- 1.2.Time Domain Perspective -- 1.2.1.Representations in the Time Domain -- 1.3.Spectral Domain Perspective -- 1.3.1.Spectral Density -- 1.3.2.Linear Filtering -- 1.3.3.Periodogram -- 1.3.4.Spectral Representation -- 1.4.Integral Representations Heuristics -- 1.4.1.Representations of a Gaussian Continuous-Time Process -- 1.5.A Heuristic Overview of the Next Chapter -- 1.6.Bibliographical Notes -- 2.1.Definitions of Long-Range Dependent Series -- 2.2.Relations Between the Various Definitions of Long-Range Dependence -- 2.2.1.Some Useful Properties of Slowly and Regularly Varying Functions -- 2.2.2.Comparing Conditions II and III -- 2.2.3.Comparing Conditions II and V -- 2.2.4.Comparing Conditions I and II -- 2.2.5.Comparing Conditions II and IV -- 2.2.6.Comparing Conditions I and IV -- 2.2.7.Comparing Conditions IV and III -- 2.2.8.Comparing Conditions IV and V -- 2.3.Short-Range Dependent Series and their Several Examples -- 2.4.Examples of Long-Range Dependent Series: FARIMA Models -- 2.4.1.FARIMA(0, d, 0) Series -- 2.4.2.FARIMA(p, d, q) Series -- 2.5.Definition and Basic Properties of Self-Similar Processes -- 2.6.Examples of Self-Similar Processes -- 2.6.1.Fractional Brownian Motion -- 2.6.2.Bifractional Brownian Motion -- 2.6.3.The Rosenblatt Process -- 2.6.4.SalphaS Levy Motion -- 2.6.5.Linear Fractional Stable Motion -- 2.6.6.Log-Fractional Stable Motion -- 2.6.7.The Telecom Process -- 2.6.8.Linear Fractional Levy Motion -- 2.7.The Lamperti Transformation -- 2.8.Connections Between Long-Range Dependent Series and Self-Similar Processes -- 2.9.Long- and Short-Range Dependent Series with Infinite Variance -- 2.9.1.First Definition of LRD Under Heavy Tails: Condition A -- 2.9.2.Second Definition of LRD Under Heavy Tails: Condition B -- 2.9.3.Third Definition of LRD Under Heavy Tails: Codifference -- 2.10.Heuristic Methods of Estimation -- 2.10.1.The R/S Method -- 2.10.2.Aggregated Variance Method -- 2.10.3.Regression in the Spectral Domain -- 2.10.4.Wavelet-Based Estimation -- 2.11.Generation of Gaussian Long- and Short-Range Dependent Series -- 2.11.1.Using Cholesky Decomposition -- 2.11.2.Using Circulant Matrix Embedding -- 2.12.Exercises -- 2.13.Bibliographical Notes -- 3.1.Aggregation of Short-Range Dependent Series -- 3.2.Mixture of Correlated Random Walks -- 3.3.Infinite Source Poisson Model with Heavy Tails -- 3.3.1.Model Formulation -- 3.3.2.Workload Process and its Basic Properties -- 3.3.3.Input Rate Regimes -- 3.3.4.Limiting Behavior of the Scaled Workload Process -- 3.4.Power-Law Shot Noise Model -- 3.5.Hierarchical Model -- 3.6.Regime Switching -- 3.7.Elastic Collision of Particles -- 3.8.Motion of a Tagged Particle in a Simple Symmetric Exclusion Model -- 3.9.Power-Law Polya's Urn -- 3.10.Random Walk in Random Scenery -- 3.11.Two-Dimensional Ising Model -- 3.11.1.Model Formulation and Result -- 3.11.2.Correlations, Dimers and Pfaffians -- 3.11.3.Computation of the Inverse -- 3.11.4.The Strong Szego Limit Theorem -- 3.11.5.Long-Range Dependence at Critical Temperature -- 3.12.Stochastic Heat Equation -- 3.13.The Weierstrass Function Connection -- 3.14.Exercises -- 3.15.Bibliographical Notes -- 4.1.Hermite Polynomials and Multiple Stochastic Integrals -- 4.2.Integral Representations of Hermite Processes -- 4.2.1.Integral Representation in the Time Domain -- 4.2.2.Integral Representation in the Spectral Domain -- 4.2.3.Integral Representation on an Interval -- 4.2.4.Summary -- 4.3.Moments, Cumulants and Diagram Formulae for Multiple Integrals -- 4.3.1.Diagram Formulae -- 4.3.2.Multigraphs -- 4.3.3.Relation Between Diagrams and Multigraphs -- 4.3.4.Diagram and Multigraph Formulae for Hermite Polynomials -- 4.4.Moments and Cumulants of Hermite Processes -- 4.5.Multiple Integrals of Order Two -- 4.6.The Rosenblatt Process -- 4.7.The Rosenblatt Distribution -- 4.8.CDF of the Rosenblatt Distribution -- 4.9.Generalized Hermite and Related Processes -- 4.10.Exercises -- 4.11.Bibliographical Notes -- 5.1.Nonlinear Functions of Gaussian Random Variables -- 5.2.Hermite Rank -- 5.3.Non-Central Limit Theorem -- 5.4.Central Limit Theorem -- 5.5.The Fourth Moment Condition -- 5.6.Limit Theorems in the Linear Case -- 5.6.1.Direct Approach for Entire Functions -- 5.6.2.Approach Based on Martingale Differences -- 5.7.Multivariate Limit Theorems -- 5.7.1.The SRD Case -- 5.7.2.The LRD Case -- 5.7.3.The Mixed Case -- 5.7.4.Multivariate Limits of Multilinear Processes -- 5.8.Generation of Non-Gaussian Long- and Short-Range Dependent Series -- 5.8.1.Matching a Marginal Distribution -- 5.8.2.Relationship Between Autocorrelations -- 5.8.3.Price Theorem -- 5.8.4.Matching a Targeted Autocovariance for Series with Prescribed Marginal -- 5.9.Exercises -- 5.10.Bibliographical Notes -- 6.1.Fractional Integrals and Derivatives -- 6.1.1.Fractional Integrals on an Interval -- 6.1.2.Riemann-Liouville Fractional Derivatives D on an Interval -- 6.1.3.Fractional Integrals and Derivatives on the Real Line -- 6.1.4.Marchaud Fractional Derivatives D on the Real Line -- 6.1.5.The Fourier Transform Perspective -- 6.2.Representations of Fractional Brownian Motion -- 6.2.1.Representation of FBM on an Interval -- 6.2.2.Representations of FBM on the Real Line -- 6.3.Fractional Wiener Integrals and their Deterministic Integrands -- 6.3.1.The Gaussian Space Generated by Fractional Wiener Integrals -- 6.3.2.Classes of Integrands on an Interval -- 6.3.3.Subspaces of Classes of Integrands -- 6.3.4.The Fundamental Martingale -- 6.3.5.The Deconvolution Formula -- 6.3.6.Classes of Integrands on the Real Line -- 6.3.7.Connection to the Reproducing Kernel Hilbert Space -- 6.4.Applications -- 6.4.1.Girsanov's Formula for FBM -- 6.4.2.The Prediction Formula for FBM -- 6.4.3.Elementary Linear Filtering Involving FBM -- 6.5.Exercises -- 6.6.Bibliographical Notes -- 7.1.Stochastic Integration with Random Integrands -- 7.1.1.FBM and the Semimartingale Property -- 7.1.2.Divergence Integral for FBM -- 7.1.3.Self-Integration of FBM -- 7.1.4.Ito's Formulas -- 7.2.Applications of Stochastic Integration -- 7.2.1.Stochastic Differential Equations Driven by FBM -- 7.2.2.Regularity of Laws Related to FBM -- 7.2.3.Numerical Solutions of SDEs Driven by FBM -- 7.2.4.Convergence to Normal Law Using Stein's Method -- 7.2.5.Local Time of FBM -- 7.3.Exercises -- 7.4.Bibliographical Notes -- 8.1.Karhunen-Loeve Decomposition and FBM -- 8.1.1.The Case of General Stochastic Processes -- 8.1.2.The Case of BM -- 8.1.3.The Case of FBM -- 8.2.Wavelet Expansion of FBM -- 8.2.1.Orthogonal Wavelet Bases -- 8.2.2.Fractional Wavelets -- 8.2.3.Fractional Conjugate Mirror Filters -- 8.2.4.Wavelet-Based Expansion and Simulation of FBM -- 8.3.Paley-Wiener Representation of FBM -- 8.3.1.Complex-Valued FBM and its Representations -- 8.3.2.Space La and its Orthonormal Basis -- 8.3.3.Expansion of FBM -- 8.4.Exercises -- 8.5.Bibliographical Notes -- 9.1.Fundamentals of Multidimensional Models -- 9.1.1.Basics of Matrix Analysis -- 9.1.2.Vector Setting -- 9.1.3.Spatial Setting -- 9.2.Operator Self-Similarity -- 9.3.Vector Operator Fractional Brownian Motions -- 9.3.1.Integral Representations -- 9.3.2.Time Reversible Vector OFBMs -- 9.3.3.Vector Fractional Brownian Motions -- 9.3.4.Identifiability Questions -- 9.4.Vector Long-Range Dependence -- 9.4.1.Definitions and Basic Properties -- 9.4.2.Vector FARIM A (0, D, 0) Series -- 9.4.3.Vector PGN Series -- 9.4.4.Fractional Cointegration -- 9.5.Operator Fractional Brownian Fields -- 9.5.1.M-Homogeneous Functions -- 9.5.2.Integral Representations -- 9.5.3.Special Subclasses and Examples of OFBFs -- 9.6.Spatial Long-Range Dependence -- 9.6.1.Definitions and Basic Properties -- 9.6.2.Examples -- 9.7.Exercises -- 9.8.Bibliographical Notes -- 10.1.Exact Gaussian MLE in the and Time Domain -- 10.2.Approximate MLE -- 10.2.1.Whittle Estimation in the Spectral Domain -- 10.2.2.Autoregressive Approximation -- 10.3.Model Selection and Diagnostics -- 10.4.Forecasting -- 10.5.R Packages and Case Studies -- 10.5.1.The ARFIMA Package -- 10.5.2.The FRACDIFF Package -- 10.6.Local Whittle Estimation -- 10.6.1.Local Whittle Estimator -- 10.6.2.Bandwidth Selection -- 10.6.3.Bias Reduction and Rate Optimality -- 10.7.Broadband Whittle Approach -- 10.8.Exercises -- 10.9.Bibliographical Notes -- A1.Fourier Series and Fourier Transforms -- A1.1.Fourier Series and Fourier Transform for Sequences -- A1.2.Fourier Transform for Functions -- A2.Fourier Series of Regularly Varying Sequences -- A3.Weak and Vague Convergence of Measures -- A3.1.The Case of Probability Measures -- A3.2.The Case of Locally Finite Measures -- A4.Stable and Heavy-Tailed Random Variables and Series -- B1.Single Integrals with Respect to Random Measures -- B1.1.Integrals with Respect to Random Measures with Orthogonal Increments -- B1.2.Integrals with Respect to Gaussian Measures -- B1.3.Integrals with Respect to Stable Measures -- B1.4.Integrals with Respect to Poisson Measures -- B1.5.Integrals with Respect to Levy Measures -- B2.Multiple Integrals with Respect to Gaussian Measures -- C1.Isonormal Gaussian Processes -- C2.Derivative Operator -- C3.Divergence Integral -- C4.Generator of the Ornstein-Uhlenbeck Semigroup.
- Subject(s):
- ISBN:
- 9781107039469 alkaline paper hardcover
1107039460 alkaline paper hardcover - Bibliography Note:
- Includes bibliographical references and index.
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