Long-Range Dependence and Self-Similarity. c Vladas Pipiras and Murad S. Taqqu

Transcription

1 Long-Range Dependence and Self-Similarity c Vladas Pipiras and Murad S. Taqqu January 24, 2016

2 Contents Contents 2 Preface 8 List of abbreviations 10 Notation 11 1 A brief overview of times series and stochastic processes Stochastic processes and time series Gaussian stochastic processes Stationarity (of increments) Weak or second-order stationarity (of increments) Time domain perspective Representations in the time domain Spectral domain perspective Spectral density Linear filtering Periodogram Spectral representation Integral representations heuristics Representations of a Gaussian continuous-time process Basics of long-range dependence and self-similarity Definitions of long-range dependent series Relations between the various definitions of long-range dependence 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 FARIMA(0,d,0) series FARIMA(p, d, q) series

3 2.5 Definition and basic properties of self-similar processes Examples of self-similar processes 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 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 The R/S method Aggregated variance method Regression in the spectral domain Wavelet-based estimation Generation of Gaussian long- and short-range dependent series Using Cholesky decomposition Using circulant matrix embedding Exercises 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 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 Model formulation and result Correlations, dimers and Pfaffians Computation of the inverse The strong Szegö limit theorem Long-range dependence at critical temperature Stochastic heat equation

4 3.13 The Weierstrass function connection Exercises Hermite processes Hermite polynomials and multiple stochastic integrals Integral representations of Hermite processes Integral representation in the time domain Integral representation in the spectral domain Integral representation on an interval Summary Moments, cumulants and diagram formulae for multiple integrals Diagram formulae Multigraphs Relation between diagrams and multigraphs Diagram and multigraph formulae for Hermite polynomials Moments and cumulants of Hermite processes Multiple integrals of order two The Rosenblatt process The Rosenblatt distribution Generalized Hermite and related processes Exercises Non-central and central limit theorems Non-linear functions of Gaussian random variables Hermite rank Non-central limit theorem Central limit theorem Limit theorems in the linear case Direct approach for entire functions Approach based on martingale differences Multivariate limit theorems The SRD case The LRD case The mixed case Multivariate limits of multilinear processes Generation of non-gaussian long- and short-range dependent series Matching a marginal distribution Relationship between autocorrelations Price theorem Matching a targeted autocovariance for series with prescribed marginal Exercises Fractional calculus and fractional Wiener integrals Fractional integrals and derivatives Fractional integrals on an interval Riemann-Liouville fractional derivatives D on an interval Fractional integrals and derivatives on the real line Marchaud fractional derivatives D on the real line

5 6.1.5 The Fourier transform perspective Representations of fractional Brownian motion Representation of FBM on an interval Representations of FBM on the real line Fractional Wiener integrals and their deterministic integrands The Gaussian space generated by fractional Wiener integrals Classes of integrands on an interval Subspaces of classes of integrands The fundamental martingale The deconvolution formula Classes of integrands on the real line Connection to the reproducing kernel Hilbert space Applications Girsanov s formula for FBM The prediction formula for FBM Elementary linear filtering involving FBM Exercises Stochastic integration with respect to fractional Brownian motion Stochastic integration with random integrands FBM and the semimartingale property Divergence integral for FBM Self-integration of FBM Itô s formulas Applications of stochastic integration Stochastic differential equations driven by FBM Regularity of laws related to FBM Numerical solutions of SDEs driven by FBM Convergence to normal law using Stein s method Local time of FBM Exercises Series representations of FBM Karhunen-Loève decomposition and FBM The case of general stochastic processes The cases of BM and FBM Wavelet expansion of FBM Orthogonal wavelet bases Fractional wavelets Fractional conjugate mirror filters Wavelet-based expansion and simulation of FBM Paley-Wiener representation of FBM Complex-valued FBM and its representations Space L a and its orthonormal basis Expansion of FBM Exercises

6 9 Multidimensional models Fundamentals of multidimensional models Basics of matrix analysis Vector setting Spatial setting Operator self-similarity Operator fractional Brownian motions Integral representations Time reversible OFBMs Vector fractional Brownian motions Identifiability questions Vector long-range dependence Definitions and basic properties Vector FARIMA(0,D,0) series Vector FGN series Fractional cointegration Operator fractional Brownian fields M homogeneous functions Integral representations Special subclasses and examples of OFBFs Spatial long-range dependence Definitions and basic properties Examples Exercises Maximum likelihood estimation methods Exact Gaussian MLE in the time domain Approximate MLE Whittle estimation in the spectral domain Autoregressive approximation Model selection and diagnostics Forecasting R packages and case studies The ARFIMA package The FRACDIFF package Local Whittle estimation Local Whittle estimator Bandwidth selection Bias reduction and rate optimality Broadband Whittle approach Exercises Historical notes and extensions Notes to Chapter Notes to Chapter Notes to Chapter Notes to Chapter Notes to Chapter

7 11.6 Notes to Chapter Notes to Chapter Notes to Chapter Notes to Chapter Notes to Chapter Other topics A Auxiliary notions and results 515 A.1 Fourier series and Fourier transforms A.1.1 Fourier series and Fourier transform for sequences A.1.2 Fourier transform for functions A.2 Fourier series of regularly varying sequences A.3 Weak and vague convergence of measures A.3.1 The case of probability measures A.3.2 The case of locally finite measures A.4 Stable and heavy-tailed random variables and series B Integrals with respect to random measures 525 B.1 Single integrals with respect to random measures B.1.1 Integrals with respect to random measures with orthogonal increments B.1.2 Integrals with respect to Gaussian measures B.1.3 Integrals with respect to stable measures B.1.4 Integrals with respect to Poisson measures B.1.5 Integrals with respect to Lévy measures B.2 Multiple integrals with respect to Gaussian measures C Basics of Malliavin calculus 537 C.1 Isonormal Gaussian processes C.2 Derivative operator C.3 Divergence integral C.4 Generator of the Ornstein-Uhlenbeck semigroup Bibliography 544 7